A quadratic polynomial is a polynomial of the second degree. This means that no variable will have an exponent greater than 2. Because this is a second-degree polynomial, it will have two real roots, or solutions. For example, x2+8x−20{\displaystyle x^{2}+8x-20} is a quadratic polynomial, because the variable x{\displaystyle x} has an exponent of 2{\displaystyle 2}. This means that the term with the exponent of 2{\displaystyle 2} is listed first, followed by the first-degree term, followed by the constant. For example, you would rewrite 8x+x2−20{\displaystyle 8x+x^{2}-20} as x2+8x−20{\displaystyle x^{2}+8x-20}. This is a necessary step for solving all polynomials. For example, x2+8x−20=0{\displaystyle x^{2}+8x-20=0}. To do this, split up the first-degree term (the x{\displaystyle x} term). You are looking for two numbers whose sum is equal to the first degree coefficient, and whose product is equal to the constant.   For example, for the quadratic polynomial x2+8x−20=0{\displaystyle x^{2}+8x-20=0}, you need to find two numbers (a{\displaystyle a} and b{\displaystyle b}), where a+b=8{\displaystyle a+b=8}, and a⋅b=−20{\displaystyle a\cdot b=-20}. Since you have −20{\displaystyle -20}, you know that one of the number will be negative. You should see that 10+(−2)=8{\displaystyle 10+(-2)=8} and 10⋅(−2)=−20{\displaystyle 10\cdot (-2)=-20}. Thus, you will split up 8x{\displaystyle 8x} into 10x−2x{\displaystyle 10x-2x} and rewrite the quadratic polynomial: x2+10x−2x−20=0{\displaystyle x^{2}+10x-2x-20=0}. To do this, factor out a term common to the first two terms in the polynomial. For example, the first two terms in the polynomial x2+10x−2x−20=0{\displaystyle x^{2}+10x-2x-20=0} are x2+10x{\displaystyle x^{2}+10x}. A term common to both is x{\displaystyle x}. Thus, the factored group is x(x+10){\displaystyle x(x+10)}. To do this, factor out a term common to the second two terms in the polynomial. For example, the second two terms in the polynomial x2+10x−2x−20=0{\displaystyle x^{2}+10x-2x-20=0} are −2x−20{\displaystyle -2x-20}. A term common to both is −2{\displaystyle -2}. Thus, the factored group is −2(x+10){\displaystyle -2(x+10)}. A binomial is a two-term expression. You already have one binomial, which is the expression in parentheses for each group. This expression should be the same for each group. The second binomial is created by combining the two terms that were factored out of each group.  For example, after factoring by grouping, x2+10x−2x−20=0{\displaystyle x^{2}+10x-2x-20=0} becomes x(x+10)−2(x+10)=0{\displaystyle x(x+10)-2(x+10)=0}. The first binomial is (x+10){\displaystyle (x+10)}. The second binomial is (x−2){\displaystyle (x-2)}. So the original quadratic polynomial, x2+8x−20=0{\displaystyle x^{2}+8x-20=0} can be written as the factored expression (x+10)(x−2)=0{\displaystyle (x+10)(x-2)=0}. To do this, solve for x{\displaystyle x} in the first binomial. For example, to find the first root for (x+10)(x−2)=0{\displaystyle (x+10)(x-2)=0}, you would first set the first binomial expression to 0{\displaystyle 0} and solve for x{\displaystyle x}. Thus:x+10=0{\displaystyle x+10=0}x+10−10=0−10{\displaystyle x+10-10=0-10}x=−10{\displaystyle x=-10}So, the first root of the quadratic polynomial x2+8x−20=0{\displaystyle x^{2}+8x-20=0} is −10{\displaystyle -10}. To do this, solve for x{\displaystyle x} in the second binomial. For example, to find the second root for (x+10)(x−2)=0{\displaystyle (x+10)(x-2)=0}, you would set the second binomial expression to 0{\displaystyle 0} and solve for x{\displaystyle x}. Thus:x−2=0{\displaystyle x-2=0}x−2+2=0+2{\displaystyle x-2+2=0+2}x=2{\displaystyle x=2}So, the second root of the quadratic polynomial x2+8x−20=0{\displaystyle x^{2}+8x-20=0} is 2{\displaystyle 2}.
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