How do you write the following quadratic equation in vertex form y=a(x-p) ^2 + q by completing the square: -2x^2 + 8x - 9? What is the vertex?

Answer 1

You can also think about it like this:

Add 9 to both sides (I call this getting rid of the "problem child")

y+9=-2x² +8x

Now divide both sides by -2. This will make it easier to see how to complete the square.

(y+9)/-2 = x²+4x

Now, look at the right side and think, "What can I add to make this a perfect square trinomial?" A perfect square trinomial is a trinomial that can be rewritten as a binomial squared. For example,x²+2x+1 , can be written as (x+1)² . You can check that with factoring. In general, to find what to add to both sides, divide the coefficient with just the x (4 in this case) by 2 and square it. You actually get 4 again in this case. Add 4 to both sides.

(y+9)/-2 +4 = x²+4x +4

Now the right side can be written as a binomial squared.

(x+2)²

Multiply both sides by -2.

y+9 = -2 (x+2)²

Remember, when you have it in vertex form, y - k = a(x - h)^2

the vertex is (h,k).

Answer 2

-2x² + 8x - 9

-2(x² - 4) - 9

-2(x² - 4 + 4) - 9 + 8

-2(x - 2)² - 1