We begin with y=2x2+4x−5. We need to rewrite this into vertex form. The easiest way to do that is to complete the square. This is a very useful way to factor, and I don't want to spend too much time explaining it right now, but I'll cover it as I go.
In order to begin completing the square we need to factor the equation until the leading coefficient (meaning a from ax2+bx+c) is 1. Sometimes we get lucky and the leading coefficient is already 1, but in this case we'll need to factor. So, we factor out the 2, which leaves us with 2(x2+2x−52). Now, I realize that fraction might look scary, but don't worry about it. Fractions are no big deal, and I advise you not to convert it into decimal form. I don't want to lose any value based on rounding.
So, our next step is to take the middle component (b, or in this case 2) and divide it in half. That gives us 1. We take that value and square it. 12 is just 1. We add that to our equation, BUT we need to subtract it. IF we add 1 and then subtract 1, we haven't changed the value of the equation, which is important. We could add 523,650,320, as long as we also subtract it.
We have 2(x2+2x+1−1−52). Now, please notice how lovely x2+2x+1 looks.T hat is a perfect square, and we can combine it to this: (x+1)2. If you don't belive me, just multiply (x+1)⋅(x+1). Now, we still need to deal with −1+52. Let's combine them and see what we have to work with.
If we put it all together, we'll have 2((x+1)2−72). I want to clean this up a bit, so let's mutiply that 2 out. −72⋅2 leaves us with just −7. We are left with 2(x+1)2−7, which is in vertex form . Nice work, we're done!
