Rewriting a quadratic function from vertex form to standard form | Khan Academy

So what I have right over here is the equation of a function in vertex form. What I want to do is rewrite it so it is in standard form. So pause this video and have a go at that before we do it together.

All right, let's just remind ourselves what standard form looks like. Standard form would be a function f(x) in the form some constant a times our second degree term plus another constant b times our first degree term plus another constant c.

So how do we put this thing into this form? Well, we essentially will expand things out. So we could start with (x - 5)². We're going to multiply (x - 5) * (x - 5), then we're going to take that, multiply it by -2, and then add 7.

So first of all, what is (x - 5) * (x - 5)? Some of y'all might already have a lot of practice doing this by taking the square of binomials, but I'll give a little bit of review. We have x * x here, which is going to be x². Then you have -5 * x, which would be -5x. I could write that out as -5x, and then you have another -5 times this x, which is going to be another -5x.

Then you're going to have -5 * -5, which is + 25. So we could write this as x² - 10x + 25, and so that's what I have boxed off here. Let me put that in parenthesis, and then I'm going to multiply that by -2 and then I'm going to add 7.

So all of this is going to be equal to f(x). So what's this going to be equal to? Well, let's take our -2 and distribute it onto each of these terms. So -2 * x², we have to be very careful here; make sure we get our signs right.

Is -2x². Minus 2 times -10x, negative times a negative is a positive, so we get +20x. Then -2 times positive 25 is -50, or we're going to subtract 50 there.

And then last but not least, we want to add 7. So all of this f(x) is going to be equal to -2x² + 20x, and then if I take -50 + 7, it's -43. Or 7 - 50 is -43, and we're done. We have rewritten the function in standard form.