How to Factor a Parabola

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Factoring a parabola requires creating polynomials in the equation if there are none. Factor out a parabola to get the vertex value with instructions from a college-level math teacher in this free video on geometry.

Part of the Video Series: Geometry Tips
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Video Transcript

So how does one factor a parabola? Hi, I'm Jimmy Chang. I've been teaching College Mathematics for nine years and factoring a parabola is one of the more intensive things that you end up doing especially in Algebra and Pre-Calculus levels. But once you know the process, things are going to be pretty straightforward. Now first and foremost, you do need to be familiar once again with the standard form of the parabola. And this is what's called the standard form; y=a(x-h)squared k. Now you notice here a tells you what kind of a parabola it is. If a is positive, it opens upward, kind of like a smiley face; and if a is negative, it's downward, a frowny face. Now the vertex refers to the tip of a parabola and if you notice here, you have to go opposite the sign for h to go with the sign for k. Now if you already have a polynomial that's in this form, that's great. But if you have a function that's not in that form, you have to make it. So, and this is how what we have to do. You have to know completely the square, so here we go. Suppose you have a polynomial of the form 2x squared - 4x 1. First and foremost, you focus on these first two terms and those two terms only. Now, you need to factor, but in order to complete the square, you have to factor out this first number. So you have to factor out the 2 from these two terms. So y is equal to 2 times x squared minus 2x and then you close the parenthesis, but you leave that 1 outside. Now completing the square; completing the square says, you have to take half the number next to the x; half of -2 which in this case is -1. And what you have to do is take the -1 and then square it. Now -1 square is equal to 1 and you take that number and you add it to this particular side. But what you're actually doing is, you're not actually adding the 1, you're adding 1 times 2 which is 2. But to preserve the equality, you have to subtract 2 to keep the balance going. So as a result, you will have y = 2(x squared - 2x 1); 1 - 2 is -1and as a result you have 2 times, these factors into (x - 1 squared) - 1 and you have standard form. As you can see, the 2 represents a opening a parabola and the vertex is (-1, -1) and that's what the tip of the parabola is. So, I'm Jimmy Chang and that's how you factor a parabola.


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