Sal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola. Created by Sal Khan and Monterey Institute for Technology and Education.
I have an equation right here. It's a second degree equation. It's a quadratic. And I know its graph is going to be a parabola. Just as a review, that means it looks something like this or it looks something like that. Because the coefficient on the x squared term here is positive, I know it's going to be an upward opening parabola. And I am curious about the vertex of this parabola. And if I have an upward opening parabola, the vertex is going to be the minimum point. If I had a downward opening parabola, then the vertex would be the maximum point. So I'm really trying to find the x value. I don't know actually where this does intersect the x-axis or if it does it all. But I want to find the x value where this function takes on a minimum value. Now, there's many ways to find a vertex. Probably the easiest, there's a formula for it. And we talk about where that comes from in multiple videos, where the vertex of a parabola or the x-coordinate of the vertex of the parabola. So the x-coordinate of the vertex is just equal to negative b over 2a. And the negative b, you're just talking about the coefficient, or b is the coefficient on the first degree term, is on the coefficient on the x term. And a is the coefficient on the x squared term. So this is going to be equal to b is negative 20. So it's negative 20 over 2 times 5. Well, this is going to be equal to positive 20 over 10, which is equal to 2. And so to find the y value of the vertex, we just substitute back into the equation. The y value is going to be 5 times 2 squared minus 20 times 2 plus 15, which is equal to let's see. This is 5 times 4, which is 20, minus 40, which is negative 20, plus 15 is negative 5. So just like that, we're able to figure out the coordinate. This coordinate right over here is the point 2, negative 5. Now it's not so satisfying just to plug and chug a formula like this. And we'll see where this comes from when you look at the quadratic formula. This is the first term. It's the x value that's halfway in between the roots. So that's one way to think about it. But another way to do it, and this probably will be of more lasting help for you in your life, because you might forget this formula. It's really just try to re-manipulate this equation so you can spot its minimum point. And we're going to do that by completing the square. So let me rewrite that. And what I'll do is out of these first two terms, I'll factor out a 5, because I want to complete a square here and I'm going to leave this 15 out to the right, because I'm going to have to manipulate that as well. So it is 5 times x squared minus 4x. And then I have this 15 out here. And I want to write this as a perfect square. And we just have to remind ourselves that if I have x plus a squared, that's going to be x squared plus 2ax plus a squared. So if I want to turn something that looks like this, 2ax, into a perfect square, I just have to take half of this coefficient and square it and add it right over here in order to make it look like that. So I'm going to do that right over here. So if I take half of negative 4, that's negative 2. If I square it, that is going to be positive 4. I have to be very careful here. I can't just willy nilly add a positive 4 here. I have equality here. If they were equal before adding the 4, then they're not going to be equal after adding the 4. So I have to do proper accounting here. I either have to add 4 to both sides or I should be careful. I have to add the same amount to both sides or subtract the same amount again. Now, the reason why I was careful there is I didn't just add 4 to the right hand side of the equation. Remember, the 4 is getting multiplied by 5. I have added 20 to the right hand side of the equation. So if I want to make this balance out, if I want the equality to still be true, I either have to now add 20 to y or I have to subtract 20 from the right hand side. So I'll do that. I'll subtract 20 from the right hand side. So I added 5 times 4. If you were to distribute this, you'll see that. I could have literally, up here, said hey, I'm adding 20 and I'm subtracting 20. This is the exact same thing that I did over here. If you distribute the 5, it becomes 5x squared minus 20x plus 20 plus 15 minus 20. Exactly what's up here. The whole point of this is that now I can write this in an interesting way. I could write this as y is equal to 5 times x minus 2 squared, and then 15 minus 20 is minus 5. So the whole point of this is now to be able to inspect this. When does this equation hit a minimum value? Well, we know that this term right over here is always going to be non-negative. Or we could say it's always going to be greater than or equal to 0. This whole thing is going to hit a minimum value when this term is equal to 0 or when x equals 2. When x equals 2, we're going to hit a minimum value. And when x equals 2, what happens? Well, this whole term is 0 and y is equal to negative 5. The vertex is 2, negative 5.