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Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | It could be less than or equal. And the reason why I did that on this first example problem is because we know how to graph that. So let's graph that. I'll try to draw it a little bit neater than that. So that is my vertical axis, my y-axis. This is my x-axis right there. That is the x-axis. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | I'll try to draw it a little bit neater than that. So that is my vertical axis, my y-axis. This is my x-axis right there. That is the x-axis. And then we know the y-intercept. The y-intercept is 3, so the point 0, 3, 1, 2, 3, is on the line, and we know we have a slope of 4, which means if we go 1 in the x direction, we're going to go up 4 in the y. So 1, 2, 3, 4. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | That is the x-axis. And then we know the y-intercept. The y-intercept is 3, so the point 0, 3, 1, 2, 3, is on the line, and we know we have a slope of 4, which means if we go 1 in the x direction, we're going to go up 4 in the y. So 1, 2, 3, 4. So it's going to be right here. And that's enough to draw a line, but we could even go back in the x direction. If we go 1 back in the x direction, we're going to go down 4. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So 1, 2, 3, 4. So it's going to be right here. And that's enough to draw a line, but we could even go back in the x direction. If we go 1 back in the x direction, we're going to go down 4. 1, 2, 3, 4. So that's also going to be a point on the line. So my best attempt at drawing this line is going to look something like something. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | If we go 1 back in the x direction, we're going to go down 4. 1, 2, 3, 4. So that's also going to be a point on the line. So my best attempt at drawing this line is going to look something like something. This is the hardest part. It's going to look something like that. That is a line. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So my best attempt at drawing this line is going to look something like something. This is the hardest part. It's going to look something like that. That is a line. It should be straight. I think you get the idea. That right there is the graph of y is equal to 4x plus 3. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | That is a line. It should be straight. I think you get the idea. That right there is the graph of y is equal to 4x plus 3. So let's think about what it means to be less than. So all of these points satisfy this inequality, but we have more. This is just these points over here. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | That right there is the graph of y is equal to 4x plus 3. So let's think about what it means to be less than. So all of these points satisfy this inequality, but we have more. This is just these points over here. What about all of these where y is less than 4x plus 3? So let's think about what this means. When, let's pick up some values for x. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | This is just these points over here. What about all of these where y is less than 4x plus 3? So let's think about what this means. When, let's pick up some values for x. When x is equal to 0, what does this say? When x is equal to 0, then that means y is going to be less than 0 plus 3. y is less than 3. When x is equal to negative 1, what is this telling us? |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | When, let's pick up some values for x. When x is equal to 0, what does this say? When x is equal to 0, then that means y is going to be less than 0 plus 3. y is less than 3. When x is equal to negative 1, what is this telling us? 4 times negative 1 is negative 4, plus 3 is negative 1. y would be less than negative 1. When x is equal to 1, what is this telling us? 4 times 1 is 4, plus 3 is 7. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | When x is equal to negative 1, what is this telling us? 4 times negative 1 is negative 4, plus 3 is negative 1. y would be less than negative 1. When x is equal to 1, what is this telling us? 4 times 1 is 4, plus 3 is 7. So y is going to be less than 7. So let's at least try to plot these. So when x is equal to, let's plot this one first. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | 4 times 1 is 4, plus 3 is 7. So y is going to be less than 7. So let's at least try to plot these. So when x is equal to, let's plot this one first. When x is equal to 0, y is less than 3. So when x is equal to 0, y is less than 3. So it's all of these points here that I'm shading in in green, satisfy that right there. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So when x is equal to, let's plot this one first. When x is equal to 0, y is less than 3. So when x is equal to 0, y is less than 3. So it's all of these points here that I'm shading in in green, satisfy that right there. If I were to look at this one over here, when x is negative 1, y is less than negative 1. So y has to be all of these points down here. When x is equal to 1, y is less than 7. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So it's all of these points here that I'm shading in in green, satisfy that right there. If I were to look at this one over here, when x is negative 1, y is less than negative 1. So y has to be all of these points down here. When x is equal to 1, y is less than 7. So it's all of these points down here. And in general, you take any point x. Let's say you take this point x right there. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | When x is equal to 1, y is less than 7. So it's all of these points down here. And in general, you take any point x. Let's say you take this point x right there. If you evaluate 4x plus 3, you're going to get the point on the line. That is that x times 4 plus 3. Now, the y's that satisfy it, it could be equal to that point on the line, or it could be less than. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | Let's say you take this point x right there. If you evaluate 4x plus 3, you're going to get the point on the line. That is that x times 4 plus 3. Now, the y's that satisfy it, it could be equal to that point on the line, or it could be less than. So it's going to go below the line. So if you were to do this for all the possible x's, you would not only get all the points on this line, which we've drawn, you would get all the points below the line. So now we have graphed this inequality. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | Now, the y's that satisfy it, it could be equal to that point on the line, or it could be less than. So it's going to go below the line. So if you were to do this for all the possible x's, you would not only get all the points on this line, which we've drawn, you would get all the points below the line. So now we have graphed this inequality. It's essentially this line, 4x plus 3, with all of the area below it shaded. Now, if this was just a less than, not less than or equal sign, we would not include the actual line. And the convention to do that is to actually make the line a dashed line. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So now we have graphed this inequality. It's essentially this line, 4x plus 3, with all of the area below it shaded. Now, if this was just a less than, not less than or equal sign, we would not include the actual line. And the convention to do that is to actually make the line a dashed line. This is the situation if we were dealing with just less than 4x plus 3, because in that situation, this wouldn't apply, and we would just have that. So the line itself wouldn't have satisfied it, just the area below it. Let's do one like that. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | And the convention to do that is to actually make the line a dashed line. This is the situation if we were dealing with just less than 4x plus 3, because in that situation, this wouldn't apply, and we would just have that. So the line itself wouldn't have satisfied it, just the area below it. Let's do one like that. So let's say we have y is greater than negative x over 2 minus 6. So a good way to start, the way I like to start these problems, is to just graph this equation right here. So let me just graph, just for fun, y is equal to, this is the same thing as negative 1 half minus 6. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | Let's do one like that. So let's say we have y is greater than negative x over 2 minus 6. So a good way to start, the way I like to start these problems, is to just graph this equation right here. So let me just graph, just for fun, y is equal to, this is the same thing as negative 1 half minus 6. So if we were to graph it, that is my vertical axis, that is my horizontal axis, and our y-intercept is negative 6. So 1, 2, 3, 4, 5, 6. So that's my y-intercept. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So let me just graph, just for fun, y is equal to, this is the same thing as negative 1 half minus 6. So if we were to graph it, that is my vertical axis, that is my horizontal axis, and our y-intercept is negative 6. So 1, 2, 3, 4, 5, 6. So that's my y-intercept. And my slope is negative 1 half. Well, that should be an x there. Negative 1 half x minus 6. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So that's my y-intercept. And my slope is negative 1 half. Well, that should be an x there. Negative 1 half x minus 6. So my slope is negative 1 half, which means when I go 2 to the right, I go down 1. So if I go 2 to the right, I'm going to go down 1. So 2 to the right, down 1. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | Negative 1 half x minus 6. So my slope is negative 1 half, which means when I go 2 to the right, I go down 1. So if I go 2 to the right, I'm going to go down 1. So 2 to the right, down 1. If I go 2 to the left, if I go negative 2, I'm going to go up 1. So negative 2, up 1. So my line is going to look like this. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So 2 to the right, down 1. If I go 2 to the left, if I go negative 2, I'm going to go up 1. So negative 2, up 1. So my line is going to look like this. My line is going to look like that. That's my best attempt at drawing the line. So that's the line of y is equal to negative 1 half x minus 6. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So my line is going to look like this. My line is going to look like that. That's my best attempt at drawing the line. So that's the line of y is equal to negative 1 half x minus 6. Now, our inequality is not greater than or equal. It's just greater than negative x over 2 minus 6, or greater than negative 1 half x minus 6. So using the same logic as before, for any x, let's say that's a particular x we want to pick. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So that's the line of y is equal to negative 1 half x minus 6. Now, our inequality is not greater than or equal. It's just greater than negative x over 2 minus 6, or greater than negative 1 half x minus 6. So using the same logic as before, for any x, let's say that's a particular x we want to pick. If you evaluate negative x over 2 minus 6, you're going to get that point right there. You're going to get the point on the line. But the y's that satisfy this inequality are the y's greater than that. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | So using the same logic as before, for any x, let's say that's a particular x we want to pick. If you evaluate negative x over 2 minus 6, you're going to get that point right there. You're going to get the point on the line. But the y's that satisfy this inequality are the y's greater than that. So it's going to be not that point. In fact, you would draw an open circle there, because you can't include the point of negative 1 half x minus 6. But it's going to be all the y's greater than that. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | But the y's that satisfy this inequality are the y's greater than that. So it's going to be not that point. In fact, you would draw an open circle there, because you can't include the point of negative 1 half x minus 6. But it's going to be all the y's greater than that. So it's going to be all the y's greater than that. And that would be true for any x. You take this x, you evaluate negative 1 half or negative x over 2 minus 6, you're going to get this point over here. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | But it's going to be all the y's greater than that. So it's going to be all the y's greater than that. And that would be true for any x. You take this x, you evaluate negative 1 half or negative x over 2 minus 6, you're going to get this point over here. But the y's that satisfy it are all the y's above that. So all of the y's that satisfy this equation, or all of the coordinates that satisfy the equation, is this entire area above the line. And we're not going to include the line. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | You take this x, you evaluate negative 1 half or negative x over 2 minus 6, you're going to get this point over here. But the y's that satisfy it are all the y's above that. So all of the y's that satisfy this equation, or all of the coordinates that satisfy the equation, is this entire area above the line. And we're not going to include the line. So the convention is to make this line into a dashed line. And let me try my best to turn it into a dashed line. I'll just erase sections of the line. |
Introduction to graphing inequalities Two-variable linear inequalities Algebra I Khan Academy.mp3 | And we're not going to include the line. So the convention is to make this line into a dashed line. And let me try my best to turn it into a dashed line. I'll just erase sections of the line. And hopefully it will look dashed to you. So I'm turning that solid line into a dashed line to show that it's just a boundary, but it's not included in the coordinates that satisfy our inequality. The coordinates that satisfy our inequality are all of this yellow stuff that I'm shading above the line. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Sometimes a quadratic polynomial, or just a quadratic itself, or quadratic expression. But all it means is a second degree polynomial. So something that's going to have a variable raised to the second power. In this case, and all of the examples will do, it'll be x. So let's say I have the quadratic expression, x squared plus 10x plus 9. And I want to factor it into the product of two binomials. How do we do that? |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | In this case, and all of the examples will do, it'll be x. So let's say I have the quadratic expression, x squared plus 10x plus 9. And I want to factor it into the product of two binomials. How do we do that? Well, let's just think about what happens if we were to take x plus a and multiply that by x plus b. If we were to multiply these two things, what happens? Well, we have a little bit of experience doing this. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | How do we do that? Well, let's just think about what happens if we were to take x plus a and multiply that by x plus b. If we were to multiply these two things, what happens? Well, we have a little bit of experience doing this. This will be x times x, which is x squared, plus x times b, which is bx, plus a times x, plus a times b, plus ab. Or if we want to add these two in the middle right here, because they have the same, they're both coefficients of x, we could write this as x squared plus, I could write it as b plus a, or a plus bx, plus ab. So in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the x term, or you could say the first degree coefficient there, that's going to be the sum of our a and b. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Well, we have a little bit of experience doing this. This will be x times x, which is x squared, plus x times b, which is bx, plus a times x, plus a times b, plus ab. Or if we want to add these two in the middle right here, because they have the same, they're both coefficients of x, we could write this as x squared plus, I could write it as b plus a, or a plus bx, plus ab. So in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the x term, or you could say the first degree coefficient there, that's going to be the sum of our a and b. This is going to be the sum of our a and b. And then the constant term is going to be the product of our a and b. Notice, this would map to this, and this would map to this. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the x term, or you could say the first degree coefficient there, that's going to be the sum of our a and b. This is going to be the sum of our a and b. And then the constant term is going to be the product of our a and b. Notice, this would map to this, and this would map to this. And of course, this is the same thing as this. So can we somehow pattern match this to that? Is there some a and b where a plus b is equal to 10, and a times b is equal to 9? |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Notice, this would map to this, and this would map to this. And of course, this is the same thing as this. So can we somehow pattern match this to that? Is there some a and b where a plus b is equal to 10, and a times b is equal to 9? Well, let's just think about it a little bit. What are the factors of 9? What are the things that a and b could be equal to? |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Is there some a and b where a plus b is equal to 10, and a times b is equal to 9? Well, let's just think about it a little bit. What are the factors of 9? What are the things that a and b could be equal to? And we're assuming that everything is an integer, and normally when we're factoring, especially when we're beginning to factor, we're dealing with integer numbers. So what are the factors of 9? They're 1, 3, and 9. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | What are the things that a and b could be equal to? And we're assuming that everything is an integer, and normally when we're factoring, especially when we're beginning to factor, we're dealing with integer numbers. So what are the factors of 9? They're 1, 3, and 9. So this could be a 3 and a 3, or it could be a 1 and a 9. Now, if it's a 3 and a 3, then you'll have 3 plus 3. That doesn't equal 10. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | They're 1, 3, and 9. So this could be a 3 and a 3, or it could be a 1 and a 9. Now, if it's a 3 and a 3, then you'll have 3 plus 3. That doesn't equal 10. But if it's a 1 and a 9, 1 times 9 is 9. 1 plus 9 is 10. So it does work. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | That doesn't equal 10. But if it's a 1 and a 9, 1 times 9 is 9. 1 plus 9 is 10. So it does work. So a could be equal to 1, and b could be equal to 9. So we could factor this as being x plus 1 times x plus 9. And if you multiply these two out using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x plus 9. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So it does work. So a could be equal to 1, and b could be equal to 9. So we could factor this as being x plus 1 times x plus 9. And if you multiply these two out using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x plus 9. So when you see something like this, when the coefficient on the x squared term, or the leading coefficient on this quadratic, is a 1, you can just say, all right, what two numbers add up to this coefficient right here? And what two numbers add up? And those same two numbers, when you take their product, have to be equal to 9. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And if you multiply these two out using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x plus 9. So when you see something like this, when the coefficient on the x squared term, or the leading coefficient on this quadratic, is a 1, you can just say, all right, what two numbers add up to this coefficient right here? And what two numbers add up? And those same two numbers, when you take their product, have to be equal to 9. And of course, this has to be in standard form. Or if it's not in standard form, you should put it in that form so that you can always say, OK, whatever's on the first degree coefficient, my a and b have to add to that. Whatever's my constant term, my a times b, the product has to be that. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And those same two numbers, when you take their product, have to be equal to 9. And of course, this has to be in standard form. Or if it's not in standard form, you should put it in that form so that you can always say, OK, whatever's on the first degree coefficient, my a and b have to add to that. Whatever's my constant term, my a times b, the product has to be that. Let's do several more examples. And I think the more examples we do, the more sense this will make. Let's say we had x squared plus 10x plus, well, I already did 10x. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Whatever's my constant term, my a times b, the product has to be that. Let's do several more examples. And I think the more examples we do, the more sense this will make. Let's say we had x squared plus 10x plus, well, I already did 10x. Let's do a different number. x squared plus 15x plus 50. We want to factor this. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Let's say we had x squared plus 10x plus, well, I already did 10x. Let's do a different number. x squared plus 15x plus 50. We want to factor this. Well, same drill. We have an x squared term, x squared term. We have a first degree term. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | We want to factor this. Well, same drill. We have an x squared term, x squared term. We have a first degree term. This should be, this right here, should be the sum of two numbers, and then this term, the constant term right here, should be the product of two numbers. So we need to think of two numbers that when I multiply them, I get 50, and when I add them, I get 15. And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | We have a first degree term. This should be, this right here, should be the sum of two numbers, and then this term, the constant term right here, should be the product of two numbers. So we need to think of two numbers that when I multiply them, I get 50, and when I add them, I get 15. And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. So what could a and b be? Let's think about the factors of 50. It could be 1 times 50, 2 times 25. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. So what could a and b be? Let's think about the factors of 50. It could be 1 times 50, 2 times 25. See, 4 doesn't go into 50. It could be 5 times 10. I think that's all of them. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | It could be 1 times 50, 2 times 25. See, 4 doesn't go into 50. It could be 5 times 10. I think that's all of them. Let's try out these numbers and see if any of these add up to 15. So 1 plus 50 does not add up to 15. 2 plus 25 does not add up to 15. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | I think that's all of them. Let's try out these numbers and see if any of these add up to 15. So 1 plus 50 does not add up to 15. 2 plus 25 does not add up to 15. But 5 plus 10 does add up to 15. So this could be 5 plus 10, and this could be 5 times 10. So if we were to factor this, this would be equal to x plus 5 times x plus 10. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | 2 plus 25 does not add up to 15. But 5 plus 10 does add up to 15. So this could be 5 plus 10, and this could be 5 times 10. So if we were to factor this, this would be equal to x plus 5 times x plus 10. And multiply it out. I encourage you to multiply this out and see that this is indeed x squared plus 15x plus 10. In fact, let's do it. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So if we were to factor this, this would be equal to x plus 5 times x plus 10. And multiply it out. I encourage you to multiply this out and see that this is indeed x squared plus 15x plus 10. In fact, let's do it. x times x, x squared. x times 10 plus 10x. 5 times x plus 5x. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | In fact, let's do it. x times x, x squared. x times 10 plus 10x. 5 times x plus 5x. 5 times 10 plus 50. Notice the 5 times 10 gave us the 50. The 5x plus the 10x is giving us the 15x in between. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | 5 times x plus 5x. 5 times 10 plus 50. Notice the 5 times 10 gave us the 50. The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x plus 50. x squared plus 15x plus 50. Let's up the stakes a little bit, introduce some negative signs in here. Let's say I had x squared minus 11x plus 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x plus 50. x squared plus 15x plus 50. Let's up the stakes a little bit, introduce some negative signs in here. Let's say I had x squared minus 11x plus 24. Now, it's the exact same principle. I need to think of two numbers that when I add them need to be equal to negative 11. a plus b need to be equal to negative 11. And a times b need to be equal to 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Let's say I had x squared minus 11x plus 24. Now, it's the exact same principle. I need to think of two numbers that when I add them need to be equal to negative 11. a plus b need to be equal to negative 11. And a times b need to be equal to 24. Now, there's something for you to think about. If when I multiply both of these numbers, I'm getting a positive number. I'm getting a 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And a times b need to be equal to 24. Now, there's something for you to think about. If when I multiply both of these numbers, I'm getting a positive number. I'm getting a 24. That means that both of these need to be positive or both of these need to be negative. That's the only way I'm going to get a positive number here. Now, if when I add them I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | I'm getting a 24. That means that both of these need to be positive or both of these need to be negative. That's the only way I'm going to get a positive number here. Now, if when I add them I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number. So the fact that their sum is negative and the fact that their product is positive tells me that both a and b are negative. a and b have to be negative. Remember, one can't be negative and the other one can't be positive because then the product would be negative. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Now, if when I add them I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number. So the fact that their sum is negative and the fact that their product is positive tells me that both a and b are negative. a and b have to be negative. Remember, one can't be negative and the other one can't be positive because then the product would be negative. And they both can't be positive because then this, when you add them, it would get you a positive number. So let's just think about what a and b can be. So two negative numbers. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Remember, one can't be negative and the other one can't be positive because then the product would be negative. And they both can't be positive because then this, when you add them, it would get you a positive number. So let's just think about what a and b can be. So two negative numbers. So let's think about the factors of 24. And we'll kind of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So two negative numbers. So let's think about the factors of 24. And we'll kind of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6. Now, which of these, when I multiply these, well, obviously when I multiply 1 times 24 I get 24. When I get 2 times 12 I get 24. So we know that all of these, the products are 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6. Now, which of these, when I multiply these, well, obviously when I multiply 1 times 24 I get 24. When I get 2 times 12 I get 24. So we know that all of these, the products are 24. But which two of these, which two factors, when I add them, should I get 11? And then we could say, let's take the negative of both of those. So when you look at these, 3 and 8 jump out. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So we know that all of these, the products are 24. But which two of these, which two factors, when I add them, should I get 11? And then we could say, let's take the negative of both of those. So when you look at these, 3 and 8 jump out. 3 times 8 is equal to 24. 3 plus 8 is equal to 11. But that doesn't quite work out, right? |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So when you look at these, 3 and 8 jump out. 3 times 8 is equal to 24. 3 plus 8 is equal to 11. But that doesn't quite work out, right? Because we have a negative 11 here. But what if we did negative 3 and negative 8? Negative 3 times negative 8 is equal to positive 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | But that doesn't quite work out, right? Because we have a negative 11 here. But what if we did negative 3 and negative 8? Negative 3 times negative 8 is equal to positive 24. Negative 3 minus 11, or sorry, negative 3 plus negative 8 is equal to negative 11. So negative 3 and negative 8 work. So if we factor this, this is going to x squared minus 11x plus 24 is going to be equal to x minus 3 times x minus 8. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Negative 3 times negative 8 is equal to positive 24. Negative 3 minus 11, or sorry, negative 3 plus negative 8 is equal to negative 11. So negative 3 and negative 8 work. So if we factor this, this is going to x squared minus 11x plus 24 is going to be equal to x minus 3 times x minus 8. Let's do another one like that. Actually, let's mix it up a little bit. Let's say I had x squared plus 5x minus 14. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So if we factor this, this is going to x squared minus 11x plus 24 is going to be equal to x minus 3 times x minus 8. Let's do another one like that. Actually, let's mix it up a little bit. Let's say I had x squared plus 5x minus 14. So here we have a different situation. The product of my two numbers is negative. a times b is equal to negative 14. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Let's say I had x squared plus 5x minus 14. So here we have a different situation. The product of my two numbers is negative. a times b is equal to negative 14. My product is negative. That tells me that one of them is positive and one of them is negative. And when I add the two, a plus b, I get it being equal to 5. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | a times b is equal to negative 14. My product is negative. That tells me that one of them is positive and one of them is negative. And when I add the two, a plus b, I get it being equal to 5. So let's think about the factors of 14 and what combinations of them, when I add them, if one is positive and one is negative, or I'm really kind of taking the difference of the two, do I get 5? So if I take 1 and 14, I'm just going to try out things. Negative 1 plus 14 is negative 13. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And when I add the two, a plus b, I get it being equal to 5. So let's think about the factors of 14 and what combinations of them, when I add them, if one is positive and one is negative, or I'm really kind of taking the difference of the two, do I get 5? So if I take 1 and 14, I'm just going to try out things. Negative 1 plus 14 is negative 13. So let me write all of the combinations that I could do. And eventually your brain will just zone in on it. So you could have negative 1 plus 14 is equal to 13. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Negative 1 plus 14 is negative 13. So let me write all of the combinations that I could do. And eventually your brain will just zone in on it. So you could have negative 1 plus 14 is equal to 13. And 1 plus negative 14 is equal to negative 13. So those don't work. That doesn't equal 5. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So you could have negative 1 plus 14 is equal to 13. And 1 plus negative 14 is equal to negative 13. So those don't work. That doesn't equal 5. Now what about 2 and 7? If I do negative 2 plus 7, that is equal to 5. We're done. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | That doesn't equal 5. Now what about 2 and 7? If I do negative 2 plus 7, that is equal to 5. We're done. That worked. I mean, we could have tried 2 plus negative 7, but that would have equaled negative 5. So that wouldn't have worked. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | We're done. That worked. I mean, we could have tried 2 plus negative 7, but that would have equaled negative 5. So that wouldn't have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. So there we have it. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So that wouldn't have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. So there we have it. We know it's x minus 2 times x plus 7. That's pretty neat. Negative 2 times 7 is negative 14. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So there we have it. We know it's x minus 2 times x plus 7. That's pretty neat. Negative 2 times 7 is negative 14. Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well honed the skill. So let's say we have x squared minus x minus 56. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Negative 2 times 7 is negative 14. Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well honed the skill. So let's say we have x squared minus x minus 56. So the product of the two numbers have to be minus 56, have to be negative 56. And their difference, because 1 is going to be positive and 1 is going to be negative, their difference has to be negative 1. And the numbers that immediately jump out in my brain, and I don't know if they jump out in your brain, we just learned this in the times tables, 56 is 8 times 7. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So let's say we have x squared minus x minus 56. So the product of the two numbers have to be minus 56, have to be negative 56. And their difference, because 1 is going to be positive and 1 is going to be negative, their difference has to be negative 1. And the numbers that immediately jump out in my brain, and I don't know if they jump out in your brain, we just learned this in the times tables, 56 is 8 times 7. I mean, there's other numbers. It's also 28 times 2. It's all sorts of things. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And the numbers that immediately jump out in my brain, and I don't know if they jump out in your brain, we just learned this in the times tables, 56 is 8 times 7. I mean, there's other numbers. It's also 28 times 2. It's all sorts of things. But 8 times 7 really jumped out into my brain because they're very close to each other. We need numbers that are very close to each other. And one of these has to be positive and one of these has to be negative. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | It's all sorts of things. But 8 times 7 really jumped out into my brain because they're very close to each other. We need numbers that are very close to each other. And one of these has to be positive and one of these has to be negative. Now, the fact that when their sum is negative tells me that the larger of these two should probably be negative. So if we take negative 8 times 7, that's equal to negative 56, and then if we take negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there. So when I factor this, this is going to be x minus 8 times x plus 7. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And one of these has to be positive and one of these has to be negative. Now, the fact that when their sum is negative tells me that the larger of these two should probably be negative. So if we take negative 8 times 7, that's equal to negative 56, and then if we take negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there. So when I factor this, this is going to be x minus 8 times x plus 7. This is often one of the hardest concepts people learn in algebra because it is a bit of an art. You have to look at all of the factors here, play with the positive and negative signs, see which of those factors, when 1 is positive and 1 is negative, add up to the coefficient on the x term. But as you do more and more practice, you'll see that it'll become a bit of second nature. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So when I factor this, this is going to be x minus 8 times x plus 7. This is often one of the hardest concepts people learn in algebra because it is a bit of an art. You have to look at all of the factors here, play with the positive and negative signs, see which of those factors, when 1 is positive and 1 is negative, add up to the coefficient on the x term. But as you do more and more practice, you'll see that it'll become a bit of second nature. Now let's step up the stakes a little bit more. Let's say we had negative x squared. Everything we've done so far had a positive coefficient, a positive 1 coefficient on the x squared term. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | But as you do more and more practice, you'll see that it'll become a bit of second nature. Now let's step up the stakes a little bit more. Let's say we had negative x squared. Everything we've done so far had a positive coefficient, a positive 1 coefficient on the x squared term. But let's say we had a negative x squared minus 5x plus 24. How do we do this? Well, the easiest way I can think of doing it is factor out a negative 1, and then it becomes just like the problems we've been doing before. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Everything we've done so far had a positive coefficient, a positive 1 coefficient on the x squared term. But let's say we had a negative x squared minus 5x plus 24. How do we do this? Well, the easiest way I can think of doing it is factor out a negative 1, and then it becomes just like the problems we've been doing before. So this is the same thing as negative 1 times positive x squared plus 5x minus 24. I just factored a negative 1 out. You can multiply negative 1 times all of these, and you'll see it becomes this. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Well, the easiest way I can think of doing it is factor out a negative 1, and then it becomes just like the problems we've been doing before. So this is the same thing as negative 1 times positive x squared plus 5x minus 24. I just factored a negative 1 out. You can multiply negative 1 times all of these, and you'll see it becomes this. Or you could factor the negative 1 out and divide all of these by negative 1, and you get that right there. Now, same game as before. I need two numbers that when I take their product, I get negative 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | You can multiply negative 1 times all of these, and you'll see it becomes this. Or you could factor the negative 1 out and divide all of these by negative 1, and you get that right there. Now, same game as before. I need two numbers that when I take their product, I get negative 24. So one will be positive, one will be negative. And when I take their sum, it's going to be 5. So let's think about 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | I need two numbers that when I take their product, I get negative 24. So one will be positive, one will be negative. And when I take their sum, it's going to be 5. So let's think about 24. Is 1 and 24. Let's see, if this is negative 1 and 24, it's negative 23. Otherwise, if it's negative 1 and 24, it would be positive 23. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So let's think about 24. Is 1 and 24. Let's see, if this is negative 1 and 24, it's negative 23. Otherwise, if it's negative 1 and 24, it would be positive 23. If it was the other way around, it would be negative 23. It doesn't work. What about 2 and 12? |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Otherwise, if it's negative 1 and 24, it would be positive 23. If it was the other way around, it would be negative 23. It doesn't work. What about 2 and 12? Well, if this is negative, if the 2 is negative, remember, one of these have to be negative. If the 2 is negative, their sum would be 10. If the 12 is negative, their sum would be negative 10. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | What about 2 and 12? Well, if this is negative, if the 2 is negative, remember, one of these have to be negative. If the 2 is negative, their sum would be 10. If the 12 is negative, their sum would be negative 10. Still doesn't work. 3 and 8. If the 3 is negative, their sum will be 5. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | If the 12 is negative, their sum would be negative 10. Still doesn't work. 3 and 8. If the 3 is negative, their sum will be 5. So it works. So if we pick negative 3 and 8, negative 3 and 8 work. So if we use it because negative 3 plus 8 is 5, negative 3 times 8 is negative 24. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | If the 3 is negative, their sum will be 5. So it works. So if we pick negative 3 and 8, negative 3 and 8 work. So if we use it because negative 3 plus 8 is 5, negative 3 times 8 is negative 24. So this is going to be equal to, can't forget that negative 1 out front. And then we factor the inside. Negative 1 times x minus 3 times x plus 8. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So if we use it because negative 3 plus 8 is 5, negative 3 times 8 is negative 24. So this is going to be equal to, can't forget that negative 1 out front. And then we factor the inside. Negative 1 times x minus 3 times x plus 8. And if you really wanted to, you could multiply the negative 1 times this. You would get 3 minus x if you did, or you don't have to. Let's do one more of these. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Negative 1 times x minus 3 times x plus 8. And if you really wanted to, you could multiply the negative 1 times this. You would get 3 minus x if you did, or you don't have to. Let's do one more of these. The more practice, the better, I think. All right, let's say I had negative x squared plus 18x minus 72. So once again, I like to factor out the negative 1. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | Let's do one more of these. The more practice, the better, I think. All right, let's say I had negative x squared plus 18x minus 72. So once again, I like to factor out the negative 1. So this is equal to negative 1 times x squared minus 18x plus 72. Now we just have to think of two numbers that when I multiply them, I get positive 72. So they have to be the same sign. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So once again, I like to factor out the negative 1. So this is equal to negative 1 times x squared minus 18x plus 72. Now we just have to think of two numbers that when I multiply them, I get positive 72. So they have to be the same sign. And that makes it easier in our head, at least in my head. When I multiply them, I get positive 72. When I add them, I get negative 18. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | So they have to be the same sign. And that makes it easier in our head, at least in my head. When I multiply them, I get positive 72. When I add them, I get negative 18. So if they're the same sign and their sum is a negative number, they both must be negative. So they're both negative. And we could go through all of the factors of 72, but the one that springs up, maybe you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9 doesn't work. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | When I add them, I get negative 18. So if they're the same sign and their sum is a negative number, they both must be negative. So they're both negative. And we could go through all of the factors of 72, but the one that springs up, maybe you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9 doesn't work. That turns into 17. That was close. Let me show you that. |
More examples of factoring quadratics with a leading coefficient of 1 Algebra II Khan Academy.mp3 | And we could go through all of the factors of 72, but the one that springs up, maybe you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9 doesn't work. That turns into 17. That was close. Let me show you that. Negative 9 plus negative 8, that is equal to negative 17. Close, but no cigar. So what other ones are? |
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