2.25: Solving Quadratic Equations by Completing the Square
So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called completing the square , which is important for our work on conics later.
Completing the Square of a Binomial Expression
In the last section, we were able to use the Square Root Property to solve the equation \((y-7)^{2}=12\) because the left side was a perfect square.
\(\begin{aligned}(y-7)^{2} &=12 \\ y-7 &=\pm \sqrt{12} \\ y-7 &=\pm 2 \sqrt{3} \\ y &=7 \pm 2 \sqrt{3} \end{aligned}\)
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \((x−k)^{2}\) in order to use the Square Root Property.
\(\begin{aligned} x^{2}-10 x+25 &=18 \\(x-5)^{2} &=18 \end{aligned}\)
What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?
Let’s look at two examples to help us recognize the patterns.
\(\begin{array}{cc}{(x+9)^{2}} & {(y-7)^{2}} \\ {(x+9 )(x+9)} & {(y-7)(y-7)} \\ {x^{2}+9 x+9 x+81} & {y^{2}-7 y-7 y+49} \\ {x^{2}+18 x+81} & {y^{2}-14 y+49}\end{array}\)
We restate the patterns here for reference.
If \(a\) and \(b\) are real numbers,
\(\begin{array}{l}(a+b)^2=a^2+2 a b+b^2 \quad \quad \underbrace{(a+b)^2}_{(\text {binomial })^2}=\underbrace{a^2}_{\text {(first term) })^2}+\underbrace{2 a b}_{2 \cdot(\text { product of }}+\underbrace{b^2}_{\text {(second term) })^2} \end{array}\)
\((a-b)^2=a^2-2 a b+b^2 \quad \quad \underbrace{(a-b)^2}_{\text {(binomial) })^2}=\underbrace{a^2}_{\text {(first term) })^2}-\underbrace{2 a b}_{\substack{\text { terms) } \\ \text { tiroduct of }}}+\underbrace{b^2}_{\text {(second term) })^2}\)
We can use this pattern to “make” a perfect square.
We will start with the expression \(x^{2}+6 x\). Since there is a plus sign between the two terms, we will use the \((a+b)^{2}\) pattern, \(a^{2}+2 a b+b^{2}=(a+b)^{2}\).
We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find \(b\). But first we start with determining \(a\). Notice that the first term of \(x^{2}+6x\) is a square, \(x^{2}\). This tells us that \(a=x\).
What number, \(b\) , when multiplied with \(2x\) gives \(6x\)? It would have to be \(3\), which is \(\frac{1}{2}(6)\). So \(b=3\).
Now to complete the perfect square trinomial, we will find the last term by squaring \(b\), which is \(3^{2}=9\).
We can now factor.
So we found that adding \(9\) to \(x^{2}+6 x\) ‘completes the square’, and we write it as \((x+3)^{2}\).
- Identify \(b\), the coefficient of \(x\).
- Find \(\left(\frac{1}{2} b\right)^{2}\), the number to complete the square.
- Add the\(\left(\frac{1}{2} b\right)^{2}\) to \(x^{2}+bx\).
- Factor the perfect square trinomial, writing it as a binomial squared.
Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
- \(x^{2}-26 x\)
- \(y^{2}-9 y\)
- \(n^{2}+\frac{1}{2} n\)
Solution :
a.
| \(\begin{array}{l}{\color{Red} x^2-b x} \\ x^2-26 x\end{array}\) | |
| The coefficient of \(x\) is -26. | |
|
Find \(\left(\frac{1}{2} b\right)^{2}\).
\(\left(\frac{1}{2} \cdot(-26)\right)^{2}\)
|
|
| Add \(169\) to the binomial to complete the square. | \(x^{2}-26 x+169\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((x-13)^{2}\) |
b.
| \(\begin{array}{l}{\color{Red} x^2-b x }\\ y^2-9 y\end{array}\) | |
| The coefficient of \(y\) is \(-9\). | |
|
Find \(\left(\frac{1}{2} b\right)^{2}\).
\(\left(\frac{1}{2} \cdot(-9)\right)^{2}\)
|
|
| Add \(\frac{81}{4}\) to the binomial to complete the square. | \(y^{2}-9 y+\frac{81}{4}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \(\left(y-\frac{9}{2}\right)^{2}\) |
c.
| \(\begin{array}{l}{\color{Red} x^2+b x} \\ n^2+\frac{1}{2} n\end{array}\) | |
| The coefficient of \(n\) is \(\frac{1}{2}\). | |
|
Find \(\left(\frac{1}{2} b\right)^{2}\).
\(\left(\frac{1}{2} \cdot \frac{1}{2}\right)^{2}\)
|
|
| Add \(\frac{1}{16}\) to the binomial to complete the square. | \(n^{2}+\frac{1}{2} n+\frac{1}{16}\) |
| Rewrite as a binomial square. | \(\left(n+\frac{1}{4}\right)^{2}\) |
Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
- \(b^{2}-4 b\)
- \(n^{2}+13 n\)
- \(q^{2}-\frac{2}{3} q\)
- Answer
-
- \((b-2)^{2}\)
- \(\left(n+\frac{13}{2}\right)^{2}\)
- \(\left(q-\frac{1}{3}\right)^{2}\)
Solving Quadratic Equations of the Form \(x^{2}+bx+c=0\) by Completing the Square
In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.
For example, if we start with the equation \(x^{2}+6x=40\), and we want to complete the square on the left, we will add 9 to both sides of the equation.
| \(x^2+6 x=40\) | |
| \(x^2+6 x+\ldots=40+\) | |
| \(x^2+6 x+{\color{Red} 9}=40+{\color{Red} 9}\) | |
| Add \(9\) to both sides to complete the square. | \((x+3)^2=49\) |
Now the equation is in the form to solve using the Square Root Property ! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.
Solve by completing the square: \(x^{2}+8x=48\)
Solution :
| Step 1 : Isolate the variable terms on one side and the constant terms on the other. | This equation has all the variables on the left. | \(\begin{array}{l}{\color{red}{x^{2}+b x\quad\:\:\: c }} \\ {x^{2}+8 x=48}\end{array}\) |
| Step 2 : Find \(\left(\frac{1}{2} \cdot b\right)^{2}\), the number to complete the square. Add it to both sides of the equation. |
Take half of \(8\) and square it. \(4^{2}=16\) Add \(16\) to BOTH sides of the equation. |
\(x^{2}+8 x+\frac{}{\color{red}{\left(\frac{1}{2} \cdot 8\right)^{2}}}\color{black}{=}48 \) \(x^{2}+8 x\color{red}{+16}\color{black}{=}48\color{red}{+16}\) |
| Step 3 : Factor the perfect square trinomial as a binomial square. |
\(x^{2}+8 x+16=(x+4)^{2}\) Add the terms on the right. |
\((x+4)^{2}=64\) |
| Step 4 : Use the Square Root Property. | \(x+4=\pm \sqrt{64}\) | |
| Step 5 : Simplify the radical and then solve the two resulting equations. |
\(x+4=\pm 8\) \(\begin{aligned} x+4 &=8 \quad x+4=-8 \\ x &=4 \quad\quad\:\:\: x=-12 \end{aligned}\) |
|
| Step 6 : Check the solutions. | Put each answer in the original equation to check. Substitute \(x=4\) and \(x=-12\). |
\(\begin{array}{r}{x^{2}+8 x=48} \\ {(\color{red}{4}\color{black}{)}^{2}+8(\color{red}{4}\color{black}{)} \stackrel{?}{=} 48} \\ {16+32\stackrel{?}{=}48} \\ {48=48}\end{array}\) \(\begin{array}{r}{x^{2}+8 x=48} \\ {(\color{red}{-12}\color{black}{)}^{2}+8(\color{red}{-12}\color{black}{)}\stackrel{?}{=} 48} \\ {144-96 \stackrel{?}{=} 48} \\ {48=48}\end{array}\) |
Solve by completing the square: \(y^{2}−10y=−9\)
- Answer
-
\(y=1, y=9\)
The steps to solve a quadratic equation by completing the square are listed here.
- Isolate the variable terms on one side and the constant terms on the other.
- Find \(\left(\frac{1}{2} \cdot b\right)^{2}\), the number needed to complete the square. Add it to both sides of the equation.
- Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right
- Use the Square Root Property.
- Simplify the radical and then solve the two resulting equations.
- Check the solutions.
When we solve an equation by completing the square, the answers will not always be integers.
Solve by completing the square: \(x^{2}+4 x=-21\)
Solution :
| \(\begin{array}{c}{\color{Red} x^2+b x=\quad c} \\ x^2+4 x=-21\end{array}\) | |
|
The variable terms are on the left side. Take half of \(4\) and square it. |
\(x^2+4 x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot 4}\right)^2}=-21\) |
| \(\left(\frac{1}{2}(4)\right)^{2}=4\) | |
| Add \(4\) to both sides. | \(x^2+4 x+{\color{Red} 4}=-21+{\color{Red} 4}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((x+2)^2=-17\) |
| Use the Square Root Property. | \(x+2= \pm \sqrt{-17}\) |
| Simplifying using complex numbers. | \(x+2= \pm \sqrt{17} i\) |
| Subtract \(2\) from each side. | \(x=-2 \pm \sqrt{17} i\) |
| Rewrite to show two solutions. | \(x=-2+\sqrt{17} i, \quad x=-2-\sqrt{17} i\) |
| We leave the check to you. |
Solve by completing the square: \(y^{2}-10 y=-35\)
- Answer
-
\(y=5+\sqrt{15} i, y=5-\sqrt{15 i}\)
In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.
Solve by completing the square: \(y^{2}-18 y=-6\)
Solution :
| \(\begin{array}{c}{\color{Red} x^2-b x =\;\; c }\\ y^2-18 y=-6\end{array}\) | |
| The variable terms are on the left side. Take half of \(-18\) and square it. | |
| \(\left(\frac{1}{2}(-18)\right)^{2}=81\) | \(y^2-18 y+\frac{}{\left({\color{Red} \frac{1}{2} \cdot(-18)}\right)^2}=-6\) |
| Add \(81\) to both sides. | \(y^2-18 y+{\color{Red} 81}=-6+{\color{Red} 81}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((y-9)^2=75\) |
| Use the Square Root Property. | \(y-9= \pm \sqrt{75}\) |
| Simplify the radical. | \(y-9= \pm 5 \sqrt{3}\) |
| Solve for \(y\). | \(y=9 \pm 5 \sqrt{3}\) |
|
Check. |
\(\begin{aligned} y^2-18 y & =-6 \\ ({\color{Red} 9+5 \sqrt{3}})^2-18({\color{Red} 9+5 \sqrt{3}}) & \stackrel{?}{=}-6 \\ 81+90 \sqrt{3}+75-162+90 \sqrt{3} & \stackrel{?}{=}-6 \\ -6 & =-6 \checkmark\end{aligned}\) \(\begin{array}{r}y^2-18 y=-6 \\ ({\color{Red} 9-5 \sqrt{3}})^2-18({\color{Red} 9-5 \sqrt{3}}) \stackrel{?}{=}-6 \\ 81+90 \sqrt{3}+75-162+90 \sqrt{3} \stackrel{?}{=}-6 \\ -6=-6\end{array}\) |
Another way to check this would be to use a calculator. Evaluate \(y^{2}−18y\) for both of the solutions. The answer should be \(−6\).
Solve by completing the square: \(y^{2}+8 y=11\)
- Answer
-
\(y=-4+3 \sqrt{3}, y=-4-3 \sqrt{3}\)
We will start the next example by isolating the variable terms on the left side of the equation.
Solve by completing the square: \(x^{2}+10 x+4=15\)
Solution :
| \(x^2+10 x+4=15\) | |
| Isolate the variable terms on the left side. Subtract \(4\) to get the constant terms on the right side. | \(x^2+10 x=11\) |
| Take half of \(10\) and square it. | |
| \(\left(\frac{1}{2}(10)\right)^{2}=25\) | \(x^2-10 x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot(10)}\right)^2}=11\) |
| Add \(25\) to both sides. | \(x^2+10 x+{\color{Red} 25}=11+{\color{Red} 25}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((x+5)^2=36\) |
| Use the Square Root Property. | \(x+5= \pm \sqrt{36}\) |
| Simplify the radical. | \(x+5= \pm 6\) |
| Solve for \(x\). | \(x=-5 \pm 6\) |
| Rewrite to show two solutions. | \(x=-5+6, \quad x=-5-6\) |
| Solve the equations. | \(x=1, \quad x=-11\) |
|
Check: |
\(\begin{aligned} x^2+10 x+4 & =15 \\ ({\color{Red} 1})^2+10({\color{Red} 1})+4 & \stackrel{?}{=} 15 \\ 1+10+4 & \stackrel{?}{=} 15 \\ 15 & =15 \checkmark\end{aligned}\) \(\begin{aligned} x^2+10 x+4 & =15 \\ ({\color{Red} -11})^2+10({\color{Red} -11})+4 & \stackrel{?}{=} 15 \\ 121+110+4 & \stackrel{?}{=} 15 \\ 15 & =15 \checkmark\end{aligned}\) |
Solve by completing the square: \(b^{2}+8 b-4=16\)
- Answer
-
\(b=-10, b=2\)
To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.
Solve by completing the square: \(n^{2}=3 n+11\)
Solution :
| \(n^2=3 n+11\) | |
| Subtract \(3n\) to get the variable terms on the left side. | \(n^2-3 n=11\) |
| Take half of \(-3\) and square it. | |
| \(\left(\frac{1}{2}(-3)\right)^{2}=\frac{9}{4}\) | \(n^2-3 n+\frac{}{\left({\color{Red} \frac{1}{2} \cdot(-3)}\right)^2}=11\) |
| Add \(\frac{9}{4}\) to both sides. | \(n^2-3 n+{\color{Red} \frac{9}{4}}=11+{\color{Red} \frac{9}{4}}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \(\left(n-\frac{3}{2}\right)^2=\frac{44}{4}+\frac{9}{4}\) |
| Add the fractions on the right side. | \(\left(n-\frac{3}{2}\right)^2=\frac{53}{4}\) |
| Use the Square Root Property. | \(n-\frac{3}{2}= \pm \sqrt{\frac{53}{4}}\) |
| Simplify the radical. | \(n-\frac{3}{2}= \pm \frac{\sqrt{53}}{2}\) |
| Solve for \(n\). | \(n=\frac{3}{2} \pm \frac{\sqrt{53}}{2}\) |
| Rewrite to show two solutions. | \(n=\frac{3}{2}+\frac{\sqrt{53}}{2}, \quad n=\frac{3}{2}-\frac{\sqrt{53}}{2}\) |
|
Check: We leave the check for you! |
Solve by completing the square: \(q^{2}=7 q-3\)
- Answer
-
\(q=\frac{7}{2}+\frac{\sqrt{37}}{2}, q=\frac{7}{2}-\frac{\sqrt{37}}{2}\)
Notice that the left side of the next equation is in factored form. But the right side is not zero. So, we cannot use the Zero Product Property since it says “If \(a⋅b=0\), then \(a=0\) or \(b=0\).” Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.
Solve by completing the square: \((x-3)(x+5)=9\)
Solution :
| \((x-3)(x+5)=9\) | |
| We multiple the binomials on the left. | \(x^2+2 x-15=9\) |
| Add \(15\) to isolate the constant terms on the right. | \(x^2+2 x \quad=24\) |
| Take half of \(2\) and square it. | |
| \(\left(\frac{1}{2} \cdot(2)\right)^{2}=1\) | \(x^2+2 x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot(2)}\right)^2}=24\) |
| Add \(1\) to both sides. | \(x^2+2 x+{\color{Red} 1}=24+{\color{Red} 1}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((x+1)^2=25\) |
| Use the Square Root Property. | \(x+1= \pm \sqrt{25}\) |
| Solve for \(x\). | \(x=-1 \pm 5\) |
| Rewrite to show two solutions. | \(x=-1+5, x=-1-5\) |
| Simplify. | \(x=4, \quad x=-6\) |
|
Check: We leave the check for you! |
Solve by completing the square: \((c-2)(c+8)=11\)
- Answer
-
\(c=-9, c=3\)
Solving Quadratic Equations of the Form \(ax^{2}+bx+c=0\) by Completing the Square
The process of completing the square works best when the coefficient of \(x^{2}\) is \(1\), so the left side of the equation is of the form \(x^{2}+bx+c\). If the \(x^{2}\) term has a coefficient other than \(1\), we take some preliminary steps to make the coefficient equal to \(1\).
Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.
Solve by completing the square: \(3 x^{2}-12 x-15=0\)
Solution :
To complete the square, we need the coefficient of \(x^{2}\) to be one. If we factor out the coefficient of \(x^{2}\) as a common factor, we can continue with solving the equation by completing the square.
| \(3 x^2-12 x-15=0\) | |
| Factor out the greatest common factor. | \(3\left(x^2-4 x-5\right)=0\) |
| Divide both sides by \(3\) to isolate the trinomial with coefficient \(1\). | \(\frac{3\left(x^2-4 x-5\right)}{3}=\frac{0}{3}\) |
| Simplify. | \(x^2-4 x-5=0\) |
| Add \(5\) to get the constant terms on the right side. | \(x^2-4 x=5\) |
| Take half of \(4\) and square it. | |
| \(\left(\frac{1}{2}(-4)\right)^{2}=4\) | \(x^2-4 x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot(4)}\right)^2}=5\) |
| Add \(4\) to both sides. | \(x^2-4 x+{\color{Red} 4}=5+{\color{Red} 4}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \((x-2)^2=9\) |
| Use the Square Root Property. | \(x-2= \pm \sqrt{9}\) |
| Solve for \(x\). | \(x-2= \pm 3\) |
| Rewrite to show two solutions. | \(x=2+3, x=2-3\) |
| Simplify. | \(x=5, \quad x=-1\) |
|
Check: |
\(\begin{aligned} x & =5 \\ 3 x^2-12 x-15 & =0 \\ 3({\color{Red} 5})^2-12({\color{Red} 5})-15 & \stackrel{?}{=} 0 \\ 75-60-15 & \stackrel{?}{=} 0 \\ 0 & =0\end{aligned}\) \(\begin{array}{l} x=-1 \\ 3 x^2-12 x-15=0 \\ 3({\color{Red} -1})^2-12({\color{Red} -1})-15 \stackrel{?}{=} 0 \\ 3+12-15 \stackrel{?}{=} 0 \\ 0=0\end{array}\) |
Solve by completing the square: \(2 m^{2}+16 m+14=0\)
- Answer
-
\(m=-7, m=-1\)
To complete the square, the coefficient of the \(x^{2}\) must be \(1\). When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.
Solve by completing the square: \(2 x^{2}-3 x=20\)
Solution :
To complete the square we need the coefficient of \(x^{2}\) to be one. We will divide both sides of the equation by the coefficient of \(x^{2}\). Then we can continue with solving the equation by completing the square.
| \(2 x^2-3 x=20\) | |
| Divide both sides by \(2\) to get the coefficient of \(x^{2}\) to be \(1\). | \(\frac{2 x^2-3 x}{2}=\frac{20}{2}\) |
| Simplify. | \(x^2-\frac{3}{2} x=10\) |
| Take half of \(-\frac{3}{2}\) and square it. | |
| \(\left(\frac{1}{2}\left(-\frac{3}{2}\right)\right)^{2}=\frac{9}{16}\) | \(x^2-\frac{3}{2} x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot\left(-\frac{3}{2}\right)}\right)^2}=10\) |
| Add \(\frac{9}{16}\) to both sides. | \(x^2-\frac{3}{2} x+{\color{Red} \frac{9}{16}}=10+{\color{Red} \frac{9}{16}}\) |
| Factor the perfect square trinomial, writing it as a binomial squared. | \(\left(x-\frac{3}{4}\right)^2=\frac{160}{16}+\frac{9}{16}\) |
| Add the fractions on the right side. | \(\left(x-\frac{3}{4}\right)^2=\frac{169}{16}\) |
| Use the Square Root Property. | \(x-\frac{3}{4}= \pm \sqrt{\frac{169}{16}}\) |
| Simplify the radical. | \(x-\frac{3}{4}= \pm \frac{13}{4}\) |
| Solve for \(x\). | \(x=\frac{3}{4} \pm \frac{13}{4}\) |
| Rewrite to show two solutions. | \(x=\frac{3}{4}+\frac{13}{4}, x=\frac{3}{4}-\frac{13}{4}\) |
| Simplify. | \(x=4, \quad x=-\frac{5}{2}\) |
|
Check: We leave the check for you! |
Solve by completing the square: \(4 t^{2}+2 t=20\)
- Answer
-
\(t=-\frac{5}{2}, t=2\)
Now that we have seen that the coefficient of \(x^{2}\) must be \(1\) for us to complete the square, we update our procedure for solving a quadratic equation by completing the square to include equations of the form \(a x^{2}+b x+c=0\).
- Divide by the leading coefficient, "a", to make the coefficient of \(x^{2}\) term \(1\).
- Isolate the variable terms on one side and the constant terms on the other.
- Find \(\left(\frac{1}{2} \cdot b\right)^{2}\), the number needed to complete the square. Add it to both sides of the equation.
- Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right
- Use the Square Root Property.
- Simplify the radical and then solve the two resulting equations.
- Check the solutions.
Solve by completing the square: \(3 x^{2}+2 x=4\)
Solution :
Again, our first step will be to make the coefficient of \(x^{2}\) one. By dividing both sides of the equation by the coefficient of \(x^{2}\), we can then continue with solving the equation by completing the square.
| \(3 x^2+2 x=4\) | |
| Divide both sides by \(3\) to make the coefficient of \(x^{2}\) equal \(1\). | \(\frac{3 x^2+2 x}{3}=\frac{4}{3}\) |
| Simplify. | \(x^2+\frac{2}{3} x=\frac{4}{3}\) |
| Take half of \(\frac{2}{3}\) and square it. | |
| \(\left(\frac{1}{2} \cdot \frac{2}{3}\right)^{2}=\frac{1}{9}\) | \(x^2+\frac{2}{3} x+\frac{}{\left({\color{Red} \frac{1}{2} \cdot \frac{2}{3}}\right)^2}=\frac{4}{3}\) |
| Add \(\frac{1}{9}\) to both sides. | \(x^2+\frac{2}{3} x+{\color{Red} \frac{1}{9}}=\frac{4}{3}+{\color{Red} \frac{1}{9}\)} |
| Factor the perfect square trinomial, writing it as a binomial squared. | \(\left(x+\frac{1}{3}\right)^2=\frac{12}{9}+\frac{1}{9}\) |
| Use the Square Root Property. | \(x+\frac{1}{3}= \pm \sqrt{\frac{13}{9}}\) |
| Simplify the radical. | \(x+\frac{1}{3}= \pm \frac{\sqrt{13}}{3}\) |
| Solve for \(x\). | \(x=-\frac{1}{3} \pm \frac{\sqrt{13}}{3}\) |
| Rewrite to show two solutions. | \(x=-\frac{1}{3}+\frac{\sqrt{13}}{3}, x=-\frac{1}{3}-\frac{\sqrt{13}}{3}\) |
|
Check: We leave the check for you! |
Solve by completing the square: \(3 y^{2}-10 y=-5\)
- Answer
-
\(y=\frac{5}{3}+\frac{\sqrt{10}}{3}, y=\frac{5}{3}-\frac{\sqrt{10}}{3}\)


3.2: Linear Independence
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A set of vectors, \(\{u_1,\: u_2,\cdots , u_n\}\), are said to be linearly independent if for any scalars \(c_1,\: c_2,\cdots , c_n\), the equation
\[c_1\text{u}_1+c_2\text{u}_2+\cdots +c_n\text{u}_n=0\nonumber \]
has only the solution \(c_1 = c_2 =\cdots = c_n = 0\). What this means is that one is unable to write any of the vectors \(\text{u}_1,\: \text{u}_2,\cdots , \text{u}_n\) as a linear combination of any of the other vectors. For instance, if there was a solution to the above equation with \(c_1\neq 0\), then we could solve that equation for \(\text{u}_1\) in terms of the other vectors with nonzero coefficients.
As an example consider whether the following three three-by-one column vectors are linearly independent:
\[\text{u}=\left(\begin{array}{c}1\\0\\0\end{array}\right),\quad\text{v}=\left(\begin{array}{c}0\\1\\0\end{array}\right),\quad\text{w}=\left(\begin{array}{c}2\\3\\0\end{array}\right).\nonumber \]
Indeed, they are not linearly independent, that is, they are linearly dependent , because \(\text{w}\) can be written in terms of \(\text{u}\) and \(\text{v}\). In fact, \(\text{w} = 2\text{u} + 3\text{v}\). Now consider the three three-by-one column vectors given by
\[\text{u}=\left(\begin{array}{c}1\\0\\0\end{array}\right),\quad\text{v}=\left(\begin{array}{c}0\\1\\0\end{array}\right),\quad\text{w}=\left(\begin{array}{c}0\\0\\1\end{array}\right).\nonumber \]
These three vectors are linearly independent because you cannot write any one of these vectors as a linear combination of the other two. If we go back to our definition of linear independence, we can see that the equation
\[a\text{u}+b\text{v}+c\text{w}=\left(\begin{array}{c}a\\b\\c\end{array}\right)=\left(\begin{array}{c}0\\0\\0\end{array}\right)\nonumber \]
has as its only solution \(a=b=c=0\).


10.3: Reading- Slides
Rocks that fall to the base of a cliff make a talus slope (figure 1). Sometimes as one rock falls, it hits another rock, which hits another rock, and begins a landslide.
Landslides and avalanches are the most dramatic, sudden, and dangerous examples of earth materials moved by gravity. Landslides are sudden falls of rock, whereas avalanches are sudden falls of snow.
Landslides
Although many types of mass movements are included in the general term “landslide,” the more restrictive use of the term refers only to mass movements, where there is a distinct zone of weakness that separates the slide material from more stable underlying material. The two major types of slides are rotational slides and translational slides. Rotational slide: This is a slide in which the surface of rupture is curved concavely upward and the slide movement is roughly rotational about an axis that is parallel to the ground surface and transverse across the slide (figure 2a). Translational slide: In this type of slide, the landslide mass moves along a roughly planar surface with little rotation or backward tilting (figure 2b). A block slide is a translational slide in which the moving mass consists of a single unit or a few closely related units that move downslope as a relatively coherent mass (figure 2c).
When large amounts of rock suddenly break loose from a cliff or mountainside, they move quickly and with tremendous force (figure 2). Air trapped under the falling rocks acts as a cushion that keeps the rock from slowing down. Landslides and avalanches can move as fast as 200 to 300 km/hour.
Landslides are exceptionally destructive. Homes may be destroyed as hillsides collapse. Landslides can even bury entire villages. Landslides may create lakes when the rocky material dams a stream. If a landslide flows into a lake or bay, they can trigger a tsunami (figure 3).
Landslides often occur on steep slopes in dry or semi-arid climates. The California coastline, with its steep cliffs and years of drought punctuated by seasons of abundant rainfall, is prone to landslides. At-risk communities have developed landslide warning systems. Around San Francisco Bay, the National Weather Service and the U.S. Geological Survey use rain gauges to monitor soil moisture. If soil becomes saturated, the weather service issues a warning. Earthquakes, which may occur on California’s abundant faults, can also trigger landslides.
Hillside properties in the San Francisco Bay Area and elsewhere may be prone to damage from landslides. Geologists are studying the warning signs and progress of local landslides to help reduce risks and give people adequate warnings of these looming threats. You can learn more here.
Contributors and Attributions
- 10.5: Erosion and Deposition by Gravity. Provided by : CK-12. Located at : http://www.ck12.org/book/CK-12-Earth-Science-For-High-School/section/10.5/ . License : CC BY-NC: Attribution-NonCommercial
- Landslide Types and Processes. Provided by : USGS. Located at : http://pubs.usgs.gov/fs/2004/3072/fs-2004-3072.html . License : Public Domain: No Known Copyright


5.27: Introduction to Charts, Diagrams, and Graphic Organizers
What you’ll learn to do: Discuss how to most effectively use charts, diagrams, and other graphics in business messages and identify potential sources for these visual aids
When you communicate data, you can’t just throw a whole bunch of numbers on a page and expect that readers will understand what you want to say. Charts, tables, and graphs help communicators organize that data in a way that helps their audiences understand the story the data tells and, hopefully, interpret it correctly.
In this section, we’ll discuss the types of charts, graphs, and diagrams available to help you show off your data in ways that make it accessible to your audience. We’ll look at specific communication challenges and determine which kind of chart or graph best illustrates your message, and finally, we’ll discuss how to format your chart so that your story is easily and quickly understood.
Contributors and Attributions
- Introduction to Charts, Diagrams and Graphic Organizers. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution


13.4: Putting It All Together
Patient Scenario
Mrs. Howard is a 73-year-old woman who was recently admitted to the medical surgical floor with pneumonia. She has an underlying history of emphysema and has experienced a recent exacerbation in dyspnea during activity. This morning when being assisted to the bathroom, she reports, “I have to stop and catch my breath when walking.” Vital signs this morning indicated oxygen saturation 91% and respiratory rate 18 on room air at rest. During report it was communicated that Mrs. Howard is able to ambulate with the assistance of one but only moves short distances around the room before she needs to stop and rest.
Applying the Nursing Process
Assessment: The nurse identifies a relevant cue that the patient, diagnosed with pneumonia and a previous history of emphysema, is experiencing increased dyspnea when walking around the room that requires her to stop and rest. Vital signs at 0700 were reviewed, and it was noted that the patient’s respiratory rate was 24 with oxygen saturation level 91% on room air at rest. The nurse gathers additional assessment data while the patient is walking and discovers her respiratory rate increases to 30 and her oxygen saturation level decreases to 85% after walking for 2 minutes. Additionally, the patient stops and catches her breath after walking approximately 10 feet, causing her to limit her mobility.
The nurse reviews the patient’s chart and finds an order for “Oxygen via nasal cannula up to 5 L/min PRN to maintain oxygen saturation at 90%.” The nurse also notes a referral for physical therapy assessment and strengthening exercises.
Based on the assessment information gathered, the following nursing care plan is created for Mrs. Howard:
Nursing Diagnosis: Impaired Physical Mobility r/t activity intolerance as manifested by decreased oxygen saturation, increased respirations, and patient report of “I have to stop and catch my breath while walking.”
Overall Goal: The patient will demonstrate improvement in mobility.
SMART Expected Outcomes:
- Mrs. Howard will ambulate 50 feet in the hallway within 24 hours.
- Mrs. Howard will maintain an oxygen saturation level of 90% or higher while walking within 24 hours.
Planning and Implementing Nursing Interventions:
The nurse plans to administer oxygen to the patient via nasal cannula as needed to maintain an oxygen saturation level of 90% or higher. The nurse will teach the patient about the importance of balancing periods of activity with periods of rest and reinforce the use of pursed-lip breathing. The nurse will encourage patient ambulation and her active participation in completing ADLs. The nurse will collaborate with physical therapy to educate the patient regarding strengthening exercises and reinforce principles of progressive exercise. The nurse plans to further assess the patient’s smoking history and promote smoking cessation.
Sample Documentation
At 0800 when assisting the patient to the bathroom, the patient reported, “I have to stop and catch my breath when walking.” Vital signs at 0700 were respiratory rate 24 and oxygen saturation level 91% on room air at rest. At 0830, vital signs were reassessed while the patient was walking. Her respiratory rate increased to 30 and her oxygen saturation level decreased to 85% after 2 minutes of walking. The patient stopped to catch her breath after walking approximately 10 feet. Oxygen via nasal cannula at 1 L/min was applied to the patient before ambulating in the hallway at 1000. The patient’s oxygen saturation level dropped to 88% after one minute of walking and the oxygen was increased to 2 L/min. The patient’s oxygen saturation then remained at 90% for the remainder of the walk, and she was able to ambulate 50 feet. Pursed-lip breathing was demonstrated and reinforced during the walk. Physical therapy was contacted and an assessment scheduled for later this morning. The patient reports a smoking history of a pack per day for 50 years. She is interested in stopping smoking. A smoking cessation brochure was provided and discussed. Dr. Smith was notified of these events at 1030.
Evaluation
Within 24 hours, Mrs. Howard successfully ambulated 50 feet in the hallway while maintaining oxygen saturation level of 90%. SMART outcomes were “met.” Planned interventions will continue. SMART outcome is revised to, “ Mrs. Howard will ambulate 100 feet in the hallway within 24 hours.”


1.1: Introduction
Astronomy Laboratory 1: Sunrise, Sunset — Moonrise, Moonset
Module Introduction
Lab 1 will be completed throughout the term, requiring your work and participation over many weeks. The final submissions for this lab and the quiz will occur at the end of the term. Please, do not wait to start on this lab, as you will need to record data and take photographs in each week in order to receive a grade.
In this lab, we will explore the day-night cycle, essential knowledge to ancient civilizations as well as indirectly, our modern society. You will first read the Learning Unit below for background information about the location and movement of Earth, the seasons, the orbit of the Moon, and using data to predict the motions of Earth and the moon as well as the effects of the motions on all of us. You will then complete the three lab activities, and quiz.
Note: throughout the term, you will need to do observations outdoors. Be sure to check the weather forecasts ahead of time so that you are able to plan around possible obstructions, such as clouds. Contact your instructor ahead of time to discuss a plan of action, if needed, to work around those obstructions. (1)
Objectives
At the end of this module, students will be able to:
- Navigate the United States Naval Observatory (USNO) website and utilize the tools on the site.
- Analyze data to determine times and locations of Sunrise, Sunset, Moonrise, and Moonset.
- Record data pertaining to the rising and falling of the Sun and Moon.
- Calculate the average change in Moonrise time. (1)
Outcomes
The material in this module includes content designed to meet the following course outcomes:
- Explain and apply major concepts in astronomy including planets, satellites, stars, meteors, galaxies, and theories of the universe.
- Demonstrate knowledge of scientific method.
- Communicate scientific ideas through oral or written assignments.
- Interpret scientific models such as formulas, graphs, tables and schematics, draw inferences from them and recognize their limitations.
- Demonstrate the ability to think critically.
- Demonstrate the ability to use scientific and quantitative reasoning. (1)
Assigned Readings
Learning Unit 1
Assignments
- USNO Astronomical Applications Website Summary
- Sunrise / Sunset Activity
- Moonrise / Moonset Activity
- Lab 1 Quiz (1)
- Authored by : Florida State College at Jacksonville. License : CC BY: Attribution


8.4: Exercises
1. Let \(G\) be a group and let \(H\leq G\) have index 2. Prove that \(H\unlhd G\text{.}\)
2. Let \(G\) be an abelian group with \(N\unlhd G\text{.}\) Prove that \(G/N\) is abelian .
3. Find the following.
- \(|2\mathbb{Z}/6\mathbb{Z}|\)
- \(|H|\text{,}\) for \(H=2+\langle 6\rangle \subseteq \mathbb{Z}_{12}\)
- \(o(2+\langle 6\rangle)\) in \(\mathbb{Z}_{12}/\langle 6\rangle\)
- \(\langle fH\rangle \) in \(D_4/H\text{,}\) where \(H=\{e,r^2\}\)
- \(|(\mathbb{Z}_6\times \mathbb{Z}_8)/(\langle 3\rangle\times \langle 2\rangle)|\)
- \(|(\mathbb{Z}_{15} \times \mathbb{Z}_{24})/\langle (5,4)\rangle|\)
4. For each of the following, find a familiar group to which the given group is isomorphic. (Hint: Consider the group order, properties such as abelianness and cyclicity, group tables, orders of elements, etc.)
- \(\mathbb{Z}/14\mathbb{Z}\)
- \(3\mathbb{Z}/12\mathbb{Z}\)
- \(S_8/A_8\)
- \((\mathbb{Z}_4 \times \mathbb{Z}_{15})/(\langle 2 \rangle \times \langle 3 \rangle )\)
- \(D_4/\langle r^2 \rangle\)
5. Let \(H\unlhd G\) with index \(k\text{,}\) and let \(a\in G\text{.}\) Prove that \(a^k\in H\text{.}\)


21.6B: Factors Affecting Pulmonary Ventilation: Compliance of the Lungs
Lung compliance refers to the magnitude of change in lung volume as a result of the change in pulmonary pressure.
- Evaluate the factors involved in lung compliance
Key Points
- A low lung compliance would mean that the lungs would need a greater-than-average change in intrapleural pressure to change the volume of the lungs.
- A high lung compliance would indicate that little pressure difference in intrapleural pressure is needed to change the volume of the lungs.
- Persons with low lung compliance due to obstructive lung diseases tend to take rapid shallow breaths and sit hunched over to make exhalation less difficult.
- Persons with high lung compliance due to restrictive lung diseases tend to have difficulty expanding and deflating the lungs.
- Two factors determine lung compliance: elasticity of the lung tissue and surface tensions at air water interfaces.
- Two factors determine lung compliance – elasticity of the lung tissue and surface tensions at air water interfaces.
Key Terms
- Lung compliance : The ability of the lungs and pleural cavity to change in volume based on changes in pressure.
EXAMPLES
Low lung compliance can be the result of interstitial lung diseases resulting from the inhalation of particulate substances such as asbestos (asbestosis) and silicon (silicosis).
Compliance is the ability of lungs and pleural cavity to expand and contract based on changes in pressure. Lung compliance is defined as the volume change per unit of pressure change across the lung, and is an important indicator of lung health and function. Measurements of lung volumes differ at the same pressure between inhalation and exhalation, meaning that lung compliance differs between inhalation and exhalation. Lung compliance can either be measured as static or dynamic based on whether only volume and pressure (static) is measured or if their changes over time are measured as well (dynamic).
Compliance and Elastic Recoil of the Lung
Compliance depends on the elasticity and surface tension of the lungs. Compliance is inversely related to the elastic recoil of the lungs, so thickening of lung tissue will decrease lung compliance. The lungs must also be able to overcome the force of surface tension from water on lung tissue during inflation in order to be compliant, and greater surface tension causes lower lung compliance. Therefore, surfactant secreted by type II epithelial cells increases lung compliance by reducing the force of surface tension.
A low lung compliance means that the lungs are “stiff” and have a higher than normal level of elastic recoil. A stiff lung would need a greater-than-average change in pleural pressure to change the volume of the lungs, and breathing becomes more difficult as a result. Low lung compliance is commonly seen in people with restrictive lung diseases, such as pulmonary fibrosis, in which scar tissue deposits in the lung making it much more difficult for the lungs to expand and deflate, and gas exchange is impaired. Pulmonary fibrosis is caused by many different types of inhalation exposures, such as silica dust.
Pulmonary Fibrosis : Pulmonary fibrosis stiffens the lungs through deposits of scar tissue, decreasing low compliance and making it more difficult for the lungs to inflate and deflate.
A high lung compliance means that the lungs are too pliable and have a lower than normal level of elastic recoil. This indicates that little pressure difference in pleural pressure is needed to change the volume of the lungs. Exhalation of air also becomes much more difficult because the loss of elastic recoil reduces the passive ability of the lungs to deflate during exhalation. High lung compliance is commonly seen in those with obstructive diseases, such of emphysema, in which destruction of the elastic tissue of the lungs from cigarette smoke exposure causes a loss of elastic recoil of the lung. Those with emphysema have considerable difficulty with exhaling breaths and tend to take fast shallow breaths and tend to sit in a hunched-over position in order to make exhalation easier.


13.2: Chapter 2
Answer Key to Chapter 2 Learning Activities
1.
| Vital Signs | |
|---|---|
| Temperature | 98.9 °F (37.2°C) |
| Heart Rate | *182 beats/min* |
| Respirations | *36 breaths/min* |
| Blood Pressure | 152/90 mm Hg |
| Oxygen Saturation | *88% on room air* |
| Capillary Refill Time | *>3* |
| Pain | 9/10 chest discomfort |
| Physical Assessment Findings | |
| Glasgow Coma Scale Score | 14 |
|---|---|
| Level of Consciousness | Alert |
| Heart Sounds | *Irregularly regular* |
| Lungs Sounds | Clear bilaterally anterior/posterior |
| Pulses-Radial | *Rapid/bounding* |
| Pulses-Pedal | *Weak* |
| Bowel Sounds | Present and active x 4 |
| Edema | Trace bilateral lower extremities |
| Skin | Cool, clammy |
2.
| Nursing Action | Indicated | Contraindicated | Nonessential |
|---|---|---|---|
| Apply oxygen at 2 liters per nasal cannula. | X | ||
| Call imagining for a STAT lung CT. | X | ||
| Perform the National Institutes of Health (NIH) Stroke Scale Neurologic Exam. | X | ||
| Obtain a comprehensive metabolic panel (CMP). | X | ||
| Obtain a STAT EKG. | X | ||
| Raise head-of-bed to less than 10 degrees. | X | ||
| Establish patent IV access. | X | ||
| Administer potassium 20 mEq IV push STAT. | X |
Answers to interactive elements are given within the interactive element.


11.7: Review Quetions
According to Ash Maurya’s typology, Airbnb is an example of what type of business model?
- direct
- multisided
- open
- marketplace
Which startup grew from a failed podcasting company called Odeo?
- StubHub
- Groupon
What is the job to be done of a milkshake according to Christensen’s team’s customer observation research in the Link to Learning video: www.youtube.com/watch?v=m3mV...ature=youtu.be?
- employ Chick-Fil-A cows
- accompany on a long morning commute
- placate children
- satisfy hunger
What is a change, typically minor, of the current business model in an attempt to capitalize on a closely related market opportunity?
- innovation
- disruption
- iteration
- increment
A tool to identify your idealized target customer and develop a better understanding of his or her environment, behavior, concerns, and aspirations.
- jobs to be done
- business plan
- business model canvas
- empathy map
A ________ is a formal document that typically describes the business and industry, market strategies, sales potential, and competitive analysis, as well as the company’s long-term goals and objectives.
- business plan
- business model
- customer empathy map
- revenue model
Select the item that does not belong to the business model canvas.
- market size
- revenue streams
- value proposition
- customer segments
The finished story benefit formula of [end customer wants] + [specific period of time] + [address objections] is designed for what component of the lean model canvas?
- unfair advantage
- unique value proposition
- early adopters
- high concept pitch
How does the social business model canvas differ from the business model canvas and lean model canvas?
- is only used by nonprofit company
- doesn’t have customer block and investment block
- has profit block to talk about ways the company can makes more money
- has impact measure, surplus, beneficiary segments, and social and customer value propositions
Which of the following has the purpose of evaluating the team, the market for the concept, the financial viability, and possible pitfalls?
- feasibility analysis
- market analysis
- financial analysis
- management analysis
Which of the following has the purpose of forecasting revenue and expenses, projecting a financial narrative, estimating project cost, and projecting cash flow?
- feasibility analysis
- market analysis
- financial analysis
- management analysis
Which of the following identifies competitors and quantifies target customers and/or users in a specific industry?
- feasibility analysis
- market analysis
- financial analysis
- management analysis
What assesses the prowess of the management and sufficiency of resources to bring the product or idea to market?
- management analysis
- operational feasibility analysis
- financial analysis
- cash flow
What are expenses that do not vary with the level of sales, such as rent, salaries, utilities, insurance and other operating expenses?
- fixed costs
- variable costs
- balance sheet
- cash flow statements
Which of the following is a formal document that typically describes the business and industry, market strategies, sales potential, and competitive analysis, as well as the company’s long-term goals and objectives?
- business plan
- business model
- customer empathy map
- revenue model
Which of these is not the purpose of a business plan?
- thoroughly explain a business idea and how it will be carried out
- planning tool and working plan over the course of several years
- requirement to start a business
- planning for “what-if” scenarios and explore new options
Which of these components are not in a business plan?
- business description
- industry and market analysis
- financial analysis
- None of these options is correct.
Estimated profits and expenses of the business is included in which of the following?
- financial analysis
- industry and market analysis
- operation plan
- marketing plan
Who is best suited to write a business plan?
- investor
- founder(s)
- consultant
- chief financial officer


36.9: Chapter Summary
This chapter discussed the male and female reproductive systems, hormones involved in their development, and various conditions that can occur throughout a client’s lifespan. Major ovarian hormones include estrogen, progestin, FSH, LS, and GnRH. Major testicular hormones include FSH, LS, GnRH, testosterone, and androgens. Hormonal drugs are used to replace low levels of natural hormones, treat some types of cancers, prevent pregnancy, minimize symptoms of menopause, and treat debilitated clients who need tissue building and increased red blood cells. Reproductive hormones are produced naturally by the body and have complex effects. Synthetic hormones have been developed for administration when natural hormone replacement is not possible. Nurses must be aware of the intricate ways in which hormones are interrelated and affect health and illness overall.
Medications used in special situations such as infertility, pregnancy, labor, and postlabor were discussed. These include drugs that can either delay (tocolytics) or stimulate (oxytocics) uterine contractions during labor and delivery, the use of abortifacients, and fertility drugs. There are also special considerations when using drugs during pregnancy and lactation. Drugs used for menopause include hormone replacement therapy and medications that treat conditions that can develop during menopause, such as osteopenia and osteoporosis. Bisphosphonates, calcium, vitamin D, and estrogen receptor modulators (ERMs) are used to treat those conditions. Male-specific conditions treated with medications include hypogonadism, delayed puberty, erectile dysfunction, and BPH. Some testicular hormones (testosterone) and anabolic steroids are used to treat clients with debilitative conditions such as chronic illness or who may have had long-term steroid use.


14.9: For Further Exploration
Note: Resources about exoplanets are provided in The Birth of Stars and the Discovery of Planets outside the Solar System.
Articles
Meteors and Meteorites
Alper, J. “It Came from Outer Space.” Astronomy (November 2002): 36. On the analysis of organic materials in meteorites.
Beatty, J. “Catch a Fallen Star.” Sky & Telescope (August 2009): 22. On the recovery of meteorites from an impact that was seen in the sky.
Durda, D. “The Chelyabinsk Super-Meteor.” Sky & Telescope (June 2013): 24. A nice summary, with photos and eyewitness reporting.
Garcia, R., & Notkin, G. “Touching the Stars without Leaving Home.” Sky & Telescope (October 2008): 32. Hunting and collecting meteorites.
Kring, D. “Unlocking the Solar System’s Past.” Astronomy (August 2006): 32. Part of a special issue devoted to meteorites.
Rubin, A. “Secrets of Primitive Meteorites.” Scientific American (February 2013): 36. What they can teach us about the environment in which the solar system formed.
Evolution of the Solar System and Protoplanetary Disks
Jewitt, D., & Young, E. “Oceans from the Skies.” Scientific American (March 2015): 36–43. How did Earth and the other inner planets get their water after the initial hot period?
Talcott, R. “How the Solar System Came to Be.” Astronomy (November 2012): 24. On the formation period of the Sun and the planets.
Young, E. “Cloudy with a Chance of Stars.” Scientific American (February 2010): 34. On how clouds of interstellar matter turn into star systems.
Websites
Meteors and Meteorites
American Meteor Society: http://www.amsmeteors.org/ . For serious observers.
British and Irish Meteorite Society: http://www.bimsociety.org/meteorites1.shtml .
Center for Meteorite Studies: meteorites.asu.edu/meteorites. This Center at Arizona State University has information for beginners under the tab "Meteorites."
Meteorites from Mars: www2.jpl.nasa.gov/snc/. A listing and links from the Jet Propulsion Lab.
Meteors and Meteor Showers: http://www.astronomy.com/observing/observe-the-solar-system/2010/04/meteors-and-meteor-showers . From Astronomy magazine.
Meteors: www.skyandtelescope.com/observing/celestial-objects-to-watch/meteors/. A collection of articles on meteor observing from Sky & Telescope magazine.
Nine Planets Meteorites and Meteors Page: http://nineplanets.org/meteorites.html .
Some Interesting Meteorite Falls of the Last Two Centuries: http://www.icq.eps.harvard.edu/meteorites-1.html .
Evolution of the Solar System and Protoplanetary Disks
Circumstellar Disk Learning Site: http://w.astro.berkeley.edu/~kalas/disksite/index.html . By Dr. Paul Kalas.
Disk Detective Project: https://www.zooniverse.org/projects/ssilverberg/disk-detective . The WISE mission is asking the public to help them find protoplanetary disks in their infrared data.
Videos
Meteors and Meteorites
Meteorites and Meteor-wrongs: https://www.youtube.com/watch?v=VQO335Y3zXo . Video with Dr. Randy Korotev of Washington U. in St. Louis (7:05).
Rare Meteorites from London’s Natural History Museum: https://www.youtube.com/watch?v=w-Rsk-ywN44 . A tour of the meteorite collection with curator Caroline Smith (18:22). Also see a short news piece about a martian meteorite: https://www.youtube.com/watch?v=1EMR2r53f2s (2:54).
What Is a Meteor Shower (and How to Watch Them): https://www.youtube.com/watch?v=xNmgvlwInCA . Top tips for watching meteor showers from the At-Bristol Science Center (3:18).
Evolution of the Solar System and Protoplanetary Disks
Origins of the Solar System: http://www.pbs.org/wgbh/nova/space/origins-solar-system.html . Video from Nova ScienceNow narrated by Neil deGrasse Tyson (13:02).
Where Do Planets Come From?: https://www.youtube.com/watch?v=zdIJUdZWlXo . Public talk by Anjali Tripathi in March 2016 in the Center for Astrophysics Observatory Nights Series (56:14).


35.14: Religion- Saying Yes to the Colonial Encounter
In the text The Female American, Unca Eliza Winkfield guides the reader through a vivid, fast-paced, and suspenseful narrative of love, family, betrayal, survival, unity, and difference. This sounds like a synopsis for a mid-day soap opera. Unca illustrates the clashing of two cultures, specifically the two religions of the cultures. Not only does Unca utilize the difference in the religions to define clear boundaries, she also assimilates to European Christian beliefs, in pursuit of unity and love, showing that religion can save the lives of people as a whole.
Early in the story religon comes up immediately after the first encounter of the European colonists and the indigenous people. Unca writes “Men, for I see you have legs, arms, and heads as we have, look to the sun, here he pointed up to that luminary, he is our god, is he yours?” (48). Here Winkfield shows that the physical likeness of the colonist and the indigenous people holds more validity than the difference in skin pigment, but all that really matters, in determining if a new invading species is friend or foe, relies solely on what God they pray to. Just before the moment of Unca’s father’s death, he is saved from having his head cut off,
“My father, expected the same fate; but just as the executioner was about to give the stroke, a maiden, who stood by the king, and whose neck, breast, and arms, were curiously adorned with jewels, diamonds, and solid pieces of gold and silver, and who was one of the king’s daughters, stroked my father with a wand. This was the signal for deliverance; he was immediately unbound, and a covering, like the Indians wore, was put round his body, and a kind of chain, formed of long grass, round his neck, of a considerable length, one end of which the princess took hold of , and gently led him along” (49).
Now, there is a lot to unpack from this quote but let’s start with the physical description of the the “maiden” whom we soon find out is the king’s daughter. She is described as firstly a maiden, meaning she is single and ready to mingle. She is also wearing “jewels, diamonds, and solid pieces of gold and silver”, meaning every kiss does begin with K.
One thing to really focus on is the verbiage “deliverance,” this word is heavily rooted in religious context and is synonymous with the word salvation, which, in the Christian faith, is attained through accepting Jesus Christ, God’s one and only begotten son ( John 3:16), as Lord and Savior.  Am I right? Anyways, to briefly conclude, this whole interaction between the daughter and Unca’s father reads of indigenous ritual tradition, specifically with intentions of a union of marriage based off the fact that the daughter quite literally “led him along,” taking Unca’s father away to get to know more about who her future hubby really is.


3.3: Attitudes Towards Aging
Ideas about the elderly are often based upon stereotypes and depictions of older adults in the media. Older people are often shown to be helpless, forgetful, slow, have dementia, to be incontinent (unable to hold their bladder), unable to live on their own, and to be unable to engage in physical inactivity. In actuality, research shows that the majority of older people are active and very involved in life activities.
In the media, elderly people are often referred to as “cute” with younger people calling them “honey”, “dear” and “sweetheart.” These terms are often condescending and should never be used to refer to an elderly patient. Home Health Aides/Personal Care Aides should always address their patient with their last name and title, such as Mr. or Mrs., unless they request otherwise. Speak with elderly patients with respect and allow them to make their own decisions and choices as much as possible. Do not treat them like a child. Just because they may be dependent on others for their care as a child is does not mean they are children. It just means they need a little extra help.
Magazine and television advertisements focus on youth and often equate it with beauty. Numerous products advertising their use will result in a more youthful appearance or help a person live longer is nearly an obsession in many cultures (US Department of Health and Human Services, National Institutes of Health on Aging, 2011).
This leaves the impression that young=beautiful and good, while old=ugly and bad. These stereotypes result in what is called ageism. Racism as we discussed in Module Two is discrimination based on someone ‘ s race . Ageism is discrimination against someone based on their age . Ageism is harmful to older people and can result in depression, anger, loss of employment, loss of housing, and loss of emotional support.
1. This term means discrimination against someone based on their age.
a). Ageism
b). Racism
c). Sexism
2. This term means discrimination based on someone’s gender.
a). Ageism
b). Racism
c). Sexism
3. Calling elderly people names such as “honey”, “sweetie”, and “dear”, “cutie” can be condescending and shows a lack of respect for the person. True or False . ________
- Answer
-
1. A 2. C 3. True
Feedback:
1. Discrimination based on age is called ageism and is harmful to people.
2. Discrimination based on a person’s gender (male or female) is known as sexism. This is harmful to people.
3. Using words such as “honey” or “dear” to refer to an older person rather than their given or preferred name demonstrates a lack of respect. Health care workers should never do this.[/hidden-answer]
Myths of Aging
According to Mauk (2008), by the year 2030, about 20% (71 million people) of the U.S. population will be over the age of 65. As many of the patients with whom the Home Health Aide/Personal Care Aide will be working will be elderly, it is important to learn about the aging process and to avoid engaging in ageism .
Check your stereotypes about the aging process! This activity has been adapted from the Oregon Department of Human Services’ Myths and Stereotypes of Aging booklet, which can be found at:
http :// www . oregon . gov / dhs / apd – dd – training / EQC % 20 Training % 20 Documents / Myths % 20 and % 20 Stereotypes % 20 of %20 Aging . pdf
True or False ?
1. The elderly are all alike. True or False . _________
2. Most elderly people are senile or have dementia. True or False . _________
3. The elderly have no worries once they retire as they can enjoy their life. True or False . _________
4.  True or False . _________
5. Most elderly people are “set in their ways” and will not change. True or False . _________
6. Elderly people are unproductive and uncreative. True or False . _________
7. The elderly have a difficult time learning and are less intelligent than younger people. True or False . _________
8. Elderly people are grouchy and hard to get along with. True or False . _________
9. Most older people fall from time to time. True or False . _________
10. Most elderly people are incontinent (unable to control their bowels or bladder). True or False . _________
- Answer
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All of these statements are false
Feedback:
1. As we age, we actually become more different. This is due to our unique life experiences. As is any other age group, the elderly are a diverse group.
2. Dementia is not a normal part of aging. Signs of confusion and changes in mental status in older adults should be looked into immediately. They can be a sign of a urinary tract infection, dehydration, stroke, or medication interaction or side effect. Most elderly people do not have dementia. This is a common stereotype presented in the media.
3. Many elderly people have many worries as they age. They may face poverty, loss of social stature, loss of social connections, health problems, and loss of independence.
4. Sexual desire and relationships do not decrease with age. The media and society often believe that older people should not have sex. This could result in feelings of guilt on the elderly person’s part, which could result in the elderly person not having sexual relations as they might wish. Physical problems could also result in the inability to have sexual relations in the way the elderly person used to, but research shows that the majority of elderly people still desire and continue to have sexual relations.
5. While older people may be slower to change their opinion than younger people, the majority of elderly people are open to change. In fact, they face many changes due to changes in physical health, social connections, death of loved ones, and illnesses.
6. many elderly people continue to be productive members of society. Even though many elderly people have retired, some continue to work in order to meet financial obligations or to continue to remain active. Outside of work, elderly people may volunteer within their community, be involved with their families, and serve as caretaker for grandchildren while their children are at work.
7. All age groups learn at a different rate and in different ways. In fact, older people have intelligence from life experiences that younger generations do not. They can offer valued wisdom based on their own life experiences to others around them. Research shows that while we do lose brain cells, we continue to gain new ones and to build new connections within our brain. The best way to build new brain cells is to remain active and continue learning throughout the lifespan.
8. This is another stereotype often seen in the media. People who tend to be grouchy and have a hard time getting along with others when they were younger will likely continue to do so when they are older. Happiness has nothing to do with aging, and in fact, the later years can be some of the happiest times of people’s lives. They may have more freedoms than they did when they were younger and be more confident and secure in themselves than when they were younger.
9. Although fall risk does increase with age, most elderly people do not fall. If a patient falls, the cause of the fall should be investigated. The fall could be due to an infection, medication side effect, or household hazard.
10. Bladder or bowel incontinence can affect people at any age. While the risk of incontinence does increase as people age because of loss of muscle tone, people of any age can suffer from incontinence. It is a stereotype that all elderly people are incontinent. New onset of incontinence should be investigated right away as it could be a sign of a urinary tract infection, medication side effects or electrolyte imbalances.


2.1: Bisection Method
The bisection method is the easiest to numerically implement and almost always works. The main disadvantage is that convergence is slow. If the bisection method results in a computer program that runs too slow, then other faster methods may be chosen; otherwise it is a good choice of method.
We want to construct a sequence \(x_0, x_1, x_2, ...\) that converges to the root \(x = r\) that solves \(f(x) = 0\). We choose \(x_0\) and \(x_1\) such that \(x_0 < r < x_1\). We say that \(x_0\) and \(x_1\) bracket the root. With \(f(r) = 0\), we want \(f(x_0)\) and \(f(x_1)\) to be of opposite sign, so that \(f(x_0)f(x_1) < 0\). We then assign x_2 to be the midpoint of \(x_0\) and \(x_1\), that is \(x_2 = (x_0 + x_1)/2\), or
\(x_2 = x_0 + \dfrac{x_1 - x_0}{2}\).
The sign of \(f(x_2)\) can then be determined. The value of \(x_3\) is then chosen as either the midpoint of \(x_0\) and \(x_2\) or as the midpoint of \(x_2\) and \(x_1\), depending on whether \(x_0\) and \(x_2\) bracket the root, or \(x_2\) and \(x_1\) bracket the root. The root, therefore, stays bracketed at all times. The algorithm proceeds in this fashion and is typically stopped when the increment to the left side of the bracket (above, given by \((x_1 - x_0)/2)\) is smaller than some required precision.


5.1: Why It Matters- ¿Adónde fuiste de vacaciones?
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Lido Pimienta –
No Pude
Lido Pimienta is a Colombian Canadian musician, singer, and songwriter. Her music incorporates a variety of styles and influences, including traditional indigenous and Afro-Colombian musical styles, as well as contemporary synthpop and electronic music. (
More about Lido Pimienta on Wikipedia
)
A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/spansih2a/?p=200
Esteman –
Fuimos Amor
Esteban Mateus Williamson, known professionally as Esteman, is a Colombian singer and songwriter from Bogotá, Colombia. (
More about Esteman on Wikipedia
)
A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/spansih2a/?p=200
Arthur Hanlon, ChocQuibTown –
No Tuve la Culpa
ChocQuibTown is a Colombian hip-hop group. Arthur Hanlon is an American pianist, songwriter and arranger who is widely known in the Latin music realm. (Wikipedia entries on
ChocQuibTown
and
Arthur Hanlon
)
A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/spansih2a/?p=200
Jennifer Lopez –
Qué Hiciste
Jennifer Lynn Lopez, also known by her nickname J.Lo, is an American actress, singer, dancer, fashion designer, producer and businesswoman. (
Wikipedia entry on Jennifer Lopez
)
A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/spansih2a/?p=200
Julieta Venegas –
Te Vi
Julieta Venegas Percevault is an American-born Mexican singer, songwriter, instrumentalist and producer who sings pop-rock in Spanish.
A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/spansih2a/?p=200
Acceso Total –
El Niño Se Hizo Hombre De Negocios
Acceso Total is a band from Durango, Mexico that plays in the
Norteño
regional music style.
https://www.youtube.com/watch?v=fV53Re5TwCw


<IP_ADDRESS>: Flour Terms and Treatments
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Flour Terms and Treatments
In addition to types of flour, you may come across various other terms when purchasing flour. These include some terms that refer to the processing and treatment of the flour, and others outlining some of the additives that may be added during the milling and refining process.
Bleached
Bleaching and maturing agents are added to whiten and improve the baking quality quickly, making it possible to market the freshest flour. Even fine wheat flours vary in color from yellow to cream when freshly milled. At this stage, the flour produces doughs that are usually sticky and do not handle well. Flour improves with age under proper storage conditions for up to one year, both in color and quality.
Because storing flour is expensive, toward the close of the 19th century, millers began to treat freshly milled flour with oxidizing agents to bleach it and give it the handling characteristics of naturally aged flour. Under the category of maturing agents are included materials such as chlorine dioxide, chlorine gas plus a small amount of nitrosyl chloride, ammonium persulfate, and ascorbic acid. No change occurs in the nutritional value of the flour when these agents are present.
There are two classes of material used to bleach flour. A common one, an organic peroxide, reacts with the yellow pigment only and has no effect on gluten quality. Chlorine dioxide, the most widely used agent in North America, neutralizes the yellow pigment and improves the gluten quality. It does, however, destroy the tocopherols (vitamin E complex).
Enriched
Replacing naturally occurring nutrients that may have been lost during the milling process. Iron and three of the most necessary B vitamins (thiamin, riboflavin, and niacin), which are partially removed during milling, are returned to white flour by a process known as enrichment. No change occurs in taste, color, texture, baking quality, or caloric value of the flour.
Fortified
Adding a nutrient that was not present before or adding in larger quantities. Folic acid (folate) is added to enriched grains in an effort to prevent neural tube defects in children. Another example is Vitamin D fortification which we discussed previously in the milk chapter.
Pre-sifted
During the milling process, flour is sifted many times through micro-fine silk. This procedure is known as pre-sifting. The mesh size used for sifting varies from flour to flour. There are more holes per square inch for cake flour than, for example, bread flour, so a cup of cake flour has significantly more minute particles than a cup of bread flour, is liable to be denser, and weighs slightly more. Sifted flour yields more volume in baked bread than unsifted flour, simply because of the increased volume of air.
Flour Additives
A number of additives may be found in commercial flours, from agents used as dough conditioners to others that aid in the fermentation process. Why use so many additives? Many of these products are complementary – that is, they work more effectively together and the end product is as close to “ideal” as possible. Nevertheless, in some countries the number of additives allowed in flour is limited. For instance, in Germany, ascorbic acid remains the only permitted additive.
Some of the additives that are commonly added to flour include those described below.
Bromate
Until the early 1990s, bromate was added to flour because it greatly sped up the oxidation or aging of flour. In the United States, bromate is allowed in some states but banned in others (e.g., California).
Azodicarbonamide (ADA)
Approved in the United States since 1962. ADA is a fast-acting flour treatment resulting in a cohesive, dry dough that tolerates high water absorption. It is not a bleach, but because it helps produce bread with a finer texture it gives an apparently whiter crumb. It does not destroy any vitamins in the dough.
Bakers who want to know if their flours contain ADA or other chemical additives can request this information from their flour suppliers.
L-Cysteine
An amino acid, L-cysteine speeds up reactions within the dough, thus reducing or almost eliminating bulk fermentation time. In effect, it gives the baker a “no-time” dough. It improves dough elasticity and gas retention.
Ascorbic Acid
Ascorbic acid was first used as a bread improver in 1932, after it was noticed that old lemon juice added to dough gave better results because it improved gas retention and loaf volume. Essentially vitamin C (ascorbic acid) has the advantage of being safe even if too much is added to the dough, as the heat of baking destroys the vitamin component. The addition of ascorbic acid consistent with artisan bread requirements is now routine for certain flours milled in North America.
Calcium Peroxide
Calcium peroxide (not to be confused with the peroxide used for bleaching flour) is another dough-maturing agent.
Glycerides
Glycerides are multi-purpose additives used in both cake mixes and yeast doughs. They are also known as surfactants, which is a contraction for “surface-acting agents.” In bread doughs, the main function of glycerides is as a crumb-softening agent, thus retarding bread staling. Glycerides also have some dough-strengthening properties.
Sodium Stearoyl Lactylate
Approved for use in the United States since 1961, this additive improves gas retention, shortens proofing time, increases loaf volume, and works as an anti-staling agent.


3.10: Personality and Work Behavior
How can organizations foster a work environment that allows employees an opportunity to develop and grow?
Personality theories that utilize the trait approach have proven popular among investigators of employee behavior in organizations. There are several reasons for this. To begin with, trait theories focus largely on the normal, healthy adult, in contrast to psychoanalytic and other personality theories that focus largely on abnormal behavior. Trait theories identify several characteristics that describe people. Allport insisted that our understanding of individual behavior could progress only by breaking behavior patterns down into a series of elements (traits). 13 "The only thing you can do about a total personality is to send flowers to it," he once said. Hence, in the study of people at work, we may discuss an employee’s dependability, emotional stability, or cognitive complexity. These traits, when taken together, form a large mosaic that provides insight into individuals. A third reason for the popularity of trait theories in the study of organizational behavior is that the traits that are identified are measurable and tend to remain relatively stable over time. It is much easier to make comparisons among employees using these tangible qualities rather than the somewhat mystical psychoanalytic theories or the highly abstract and volatile self theories.
The number of traits people are believed to exhibit varies according to which theory we employ. In an exhaustive search, over 17,000 can be identified. Obviously, this number is so large as to make any reasonable analysis of the effects of personality in the workplace impossible. In order for us to make any sense out of this, it is necessary for us to concentrate on a small number of personality variables that have a direct impact on work behavior. If we do this, we can identify six traits that seem to be relatively important for our purposes here. It will be noted that some of these traits (for example, self-esteem or locus of control) have to do with how we see ourselves, whereas other traits (for example, introversion-extroversion or dependability) have to do with how we interact with others. Moreover, these traits are largely influenced by one’s personality development and, in turn, influence actual attitudes and behaviors at work, as shown in Figure \(\PageIndex{1}\).
Self-Esteem
One trait that has emerged recently as a key variable in determining work behavior and effectiveness is an employee’s self-esteem. Self-esteem can be defined as one’s opinion or belief about one’s self and self-worth. It is how we see ourselves as individuals. Do we have confidence in ourselves? Do we think we are successful? Attractive? Worthy of others’ respect or friendship?
Research has shown that high self-esteem in school-age children enhances assertiveness, independence, and creativity. People with high self-esteem often find it easier to give and receive affection, set higher goals for personal achievement, and exert energy to try to attain goals set for them. Moreover, individuals with high self-esteem will be more likely to seek higher-status occupations and will take more risks in the job search. For example, one study found that students possessing higher self-esteem were more highly rated by college recruiters, received more job offers, and were more satisfied with their job search than students with low self-esteem.14 Hence, personality traits such as this one can affect your job and career even before you begin work!
Locus of Control
Locus of control refers to the tendency among individuals to attribute the events affecting their lives either to their own actions or to external forces; it is a measure of how much you think you control your own destiny. Two types of individual are identified. People with an internal locus of control tend to attribute their successes—and failures—to their own abilities and efforts. Hence, a student would give herself credit for passing an examination; likewise, she would accept blame for failing.
In contrast, people with an external locus of control tend to attribute things that happen to them as being caused by someone or something else. They give themselves neither credit nor blame. Hence, passing an exam may be dismissed by saying it was “too easy,” whereas failing may be excused by convincing one’s self that the exam was “unfair.”
If you want to determine your own locus of control, fill out the self-assessment in the end-of-chapter assignments. This is an abbreviated and adapted version of the scale originally developed by Rotter. When you have finished, refer to that reference for scoring procedures.
Recent research on locus of control suggests that people with an internal locus of control (1) exhibit greater work motivation, (2) have stronger expectations that effort will lead to actual high job performance, (3) perform better on tasks requiring learning or problem-solving, (4) typically receive higher salaries and salary increases, and (5) exhibit less job-related anxiety than externals. 15 Locus of control has numerous implications for management. For example, consider what would happen if you placed an “internal” under tight supervision or an “external” under loose supervision. The results probably would not be very positive. Or what would happen if you placed both internals and externals on a merit-based compensation plan? Who would likely perform better? Who might perform better under a piece-rate system?
Introversion-Extroversion
The third personality dimension we should consider focuses on the extent to which people tend to be shy and retiring or socially gregarious. Introverts ( introversion ) tend to focus their energies inwardly and have a greater sensitivity to abstract feelings, whereas extroverts ( extroversion ) direct more of their attention to other people, objects, and events. Research evidence suggests that both types of people have a role to play in organizations. 16 Extroverts more often succeed in first-line management roles, where only superficial "people skills” are required; they also do better in field assignments—for example, as sales representatives. Introverts, on the other hand, tend to succeed in positions requiring more reflection, analysis, and sensitivity to people’s inner feelings and qualities. Such positions are included in a variety of departments within organizations, such as accounting, personnel, and computer operations. In view of the complex nature of modern organizations, both types of individuals are clearly needed.
Authoritarianism and Dogmatism
Authoritarianism refers to an individual’s orientation toward authority. More specifically, an authoritarian orientation is generally characterized by an overriding conviction that it is right and proper for there to be clear status and power differences among people. 17 According to T.W. Adorno, a high authoritarian is typically (1) demanding, directive, and controlling of her subordinates; (2) submissive and deferential toward superiors;
(3) intellectually rigid; (4) fearful of social change; (5) highly judgmental and categorical in reactions to others; (6) distrustful; and (7) hostile in response to restraint. Nonauthoritarians, on the other hand, generally believe that power and status differences should be minimized, that social change can be constructive, and that people should be more accepting and less judgmental of others.
In the workplace, the consequences of these differences can be tremendous. Research has shown, for example, that employees who are high in authoritarianism often perform better under rigid supervisory control, whereas those rated lower on this characteristic perform better under more participative supervision. 18 Can you think of other consequences that might result from these differences?
Related to this authoritarianism is the trait of dogmatism. Dogmatism refers to a particular cognitive style that is characterized by closed-mindedness and inflexibility. 19 The dimension has particularly profound implications for managerial decision-making; it is found that dogmatic managers tend to make decisions quickly, based on only limited information and with a high degree of confidence in the correctness of their decisions. 20 Do you know managers (or professors) who tend to be dogmatic? How does this behavior affect those around them?
Dependability
Finally, people can be differentiated with respect to their behavioral consistency, or dependability . Individuals who are seen as self-reliant, responsible, consistent, and dependable are typically considered to be desirable colleagues or group members who will cooperate and work steadfastly toward group goals. 21 Personnel managers often seek a wide array of information concerning dependability before hiring job applicants. Even so, contemporary managers often complain that many of today’s workers simply lack the feeling of personal responsibility necessary for efficient operations. Whether this is a result of the personal failings of the individuals or a lack of proper motivation by superiors remains to be determined.
Obviously, personality factors such as those discussed here can play a major role in determining work behavior both on the shop floor and in the executive suite. A good example of this can be seen in the events leading up to the demise of one of America’s largest and oldest architectural firms. Observe the role of personality in the events that follow.
Managing Change
Personality Clash: Design vs. Default
Philip Johnson, at age 86, was considered the dean of American architecture and was known for such landmarks as the AT&T building in New York and the Pennzoil Center in Houston, but he was also forced out of the firm that he built, only to watch it fall into default and bankruptcy.
In 1969, Johnson invited John Burgee, who was just 35, to become his sole partner to handle the management side of the business and thereby allow him to focus on the creative side. “I picked John Burgee as my righthand man. Every design architect needs a Burgee. The more leadership he took, the happier I was,” Johnson said. Burgee’s personality was perfectly suited to the nuts-and-bolts tasks of managing the firm and overseeing the projects through construction.
For all his management effort, Burgee felt that only Johnson’s name ever appeared in the press. “It was always difficult for me, being a younger man and less flamboyant,” commented Burgee. Eventually, Burgee was able to get Johnson to change the name of the firm, first to Philip Johnson & John Burgee Architects, then to Johnson/Burgee Architects, and finally to John Burgee Architects, with Philip Johnson. Although Burgee wanted to be involved in all aspects of the business, Johnson was unwilling to relinquish control over design to Burgee.
In 1988, Burgee sent a four-page memo to Johnson in which he listed each of the firm’s 24 projects and outlined the ones for which Johnson could initiate designs, initiate contact with clients, or work on independently at home. Burgee also instructed Johnson not to involve himself with the younger architects or advise them on their drawings.
The clash of the creative personality of Johnson and the controlling personality of Burgee came to a climax when Burgee asked Johnson to leave the firm. Unfortunately, Burgee underestimated the reaction of clients and lost many key contracts. Eventually, Burgee had to file for bankruptcy, and Johnson continued working on his own, including a project for Estée Lauder.
Source: Michelle Pacelle, “Design Flaw.” Wall Street Journal, September 2, 1992, p. A1, A5.
Exercise \(\PageIndex{1}\)
1. What are the things that managers can do to foster an environment where employees can gain personal development and grow?
13. G.W. Allport, Pattern and Growth in Personality (New York: Holt, Rinehart and Winston, 1961).
14. R. A. Ellis and M. S. Taylor, “Role of Self-Esteem within the Job Search Process,” Journal of Applied Psychology, 1983, 68, pp. 632–640.
15. P. Spector, “Behavior in Organizations as a Function of Locus of Control,” Psychological Bulletin, May 1982, pp. 482–497; P. Nystrom, “Managers’ Salaries and Their Beliefs About Reinforcement Control,” Journal of Social Psychology, August 1983, pp. 291–292.
16. L. R. Morris, Extroversion and Introversion: An Interactional Perspective (New York: Hemisphere, 1979), p.8.
17. T. W. Adorno, E. Frenkel-Brunswik, and D. J. Levinson, The Authoritarian Personality (New York: Harper & Row, 1950).
18. V. H. Vroom, Some Personality Determinants of the Effects of Participation (Englewood Cliffs, N.J.: Prentice- Hall, 1960).
19. M. Rokeach, The Open and Closed Mind (New York: Basic Books, 1960).
20. R. N. Taylor and M. D. Dunnette, “Influence of Dogmatism, Risk-Taking Propensity, and Intelligence on Decision-Making Strategies for a Sample of Industrial Managers,” Journal of Applied Psychology, 1974, 59, pp. 420–423.
Exhibit 2.2 (Attribution: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license)


13.2: Video- Prioritizing Marketing Communications
Given all the different marketing communication tools and opportunities out there, it can be hard to prioritize and choose where to focus your attention and marketing efforts. In this TEDx talk, Google’s Nick Scarpino provides a common-sense framework to help you make the biggest impact with whatever marketing resources are available to you.
A link to an interactive elements can be found at the bottom of this page.
Contributors and Attributions
All rights reserved content
- A Guide for Prioritizing Marketing Communications: Nick Scarpino at TEDxUofIChicago. Provided by : TEDx. Located at : https://youtu.be/UhQ2T5V2SQE?list=PLzGAKV7EcDXvAMT-rhjYDfBOTmd_2dSiQ . License : All Rights Reserved . License Terms : Standard YouTube license


15.2: Walks, trails, and paths
Suppose \(G = (V,E)\) is a graph.
a finite sequence \(v_0, e_1, v_1, e_2, \ldots, v_{n - 1}, e_n, v_n\) of elements from \(V \cup E\text{,}\) with each \(v_i \in V\) and each \(e_i \in E\text{,}\) such that edge \(e_i\) connects vertices \(v_{i-1}\) and \(v_i\)
a walk that ends at the same vertex at which it started (that is, \(v_n = v_0\))
a walk that isn't closed (that is, \(v_n \neq v_0\))
a walk that traverses no edge more than once
a walk that passes through no vertex more than once, except possibly the endpoints \(v_0,v_n\)
We may also apply the adjectives open and closed to trails and paths.
Consider the graph in Example 15.1.1 .
- The “trip” we found in the example is a trail of maximal length starting at vertex \(C\text{.}\)
- The walks \(C,r_1,R,r_4,D\) and \(C,r_1,R,r_2,C\) are both paths, the first open and the second closed.
- The walk \(C,r_1,R,r_4,D,r_4,R\) is neither a path nor a trail.
Consider the following graph.
This graph has the following properties.
- Every path or trail passing through \(v_1\) must start or end there but cannot be closed, except for the closed paths:
- \(v_1\text{;}\)
- \(v_1, e_1, v_2, e_1, v_1\text{;}\)
- \(v_2, e_1, v_1, e_1, v_2\text{;}\)
- Walk \(v_1, e_1, v_2, e_5, v_3, e_4, v_4,\) is both a trail and a path.
- Walk \(v_1, e_1, v_2, e_5, v_3, e_6, v_3, e_4, v_4,\) is a trail but not a path.
Consider again the graph in Figure \(\PageIndex{1}\) from Example \(\PageIndex{2}\). How many trails from \(v_3\) to \(v_4\) exist? How many of those trails are paths? Are there any paths from \(v_3\) to \(v_4\) that are not trails?
Solution
We can solve this using a graph! The graph in Figure \(\PageIndex{1}\) was created by mapping out all possible trails starting at \(v_3\) and ending at \(v_4\text{,}\) moving across one edge at a time. Each node in this new (directed) graph is labelled with a partial walk that is a continuation of the walk assigned to the node above it. Each leg in the graph stops when the associated walk being followed reaches \(v_4\) and cannot be continued without repeating another edge. To save space in the node labels, we have used “…” to mean the walk from the previous node.
Counting all the nodes in the graph of Figure \(\PageIndex{2}\) that are labelled with a walk that ends in \(v_4\text{,}\) we see that there are ten trails from \(v_3\) to \(v_4\text{.}\) Also, we can easily see that only three of the trails are paths.
We can use the same technique to map out all paths from \(v_3\) to \(v_4\text{,}\) but this time we terminate a leg when we cannot move off a vertex without repeating a vertex that is already visited in that walk. (Note that the walk \(v_3,e_6,v_3\) is a path, but if we extend this walk in any way it will no longer be a path.)
So there are only three paths from \(v_3\) to \(v_4\text{,}\) and each of them is a trail.
Every open path is a trail.
- Proof
-
We will prove the contrapositive: a walk that is not a trail cannot be an open path. So suppose \(W\) is a walk in a graph, and that \(W\) traverses edge \(e\) twice.
Case \(e\) is a loop.
Then \(W\) passes through the vertex incident to \(e\) at least three times, hence is not a path.Case \(e\) is not a loop.
Write \(e=\{v,v'\}\text{.}\) Initially, there are two possibilities to consider. If each of the two assumed traversals of \(e\) moves from \(v\) to \(v'\text{,}\) then \(W\) passes through each of \(v,v'\) at least twice, and hence is not a path. If the two assumed traversals of \(e\) move \(v\) to \(v'\) and \(v'\) to \(v\) respectively, then \(W\) passes through \(v\) at least twice. If \(W\) traverses \(v\) twice because it both starts and ends there, then \(W\) is not open. If \(W\) is open and traverses \(v\) twice, then \(W\) is not a path. So in any case, \(W\) is not an open path.
In Activity 15.1 , you are asked to create counterexamples of some statements related to the above proposition.