Question

What is the vertex form of `y=5x^2 - 10x - 75`?

Answer 1

`y=5(x-1)^2-80`, meaning the vertex is at the point `(x,y)=(1,-80)`.

Explanation:

First, factor out the coefficient of `x^2`, which is 5, out of the first two terms:

`y=5x^2-10x-75=5(x^2-2x)-75`.

Next, complete the square on the expression inside the parentheses. Take the coefficient of `x`, which is `-2`, divide it by 2 and square it to get `1`. Add this number inside the parentheses and compensate for this change by subtracting `5*1 = 5` outside of the parentheses as follows:

`y=5(x^2-2x+1)-75-5`.

This trick makes the expression inside the parentheses a perfect square to get the final answer:

`y=5(x-1)^2-80`.

The graph of this function is a parabola opening upward with a minimum at the vertex `(x,y)=(1,-80)`.