Question

How can you use the quadratic formula to find the vertex of a parabola?

Answer 1

Average the roots that come from the quadratic formula to see that the `x`-coordinate of the vertex of `ax^2+bx+c=0` is `x=-b/(2a)`.

Explanation:

The quadratic formula says the roots of the quadratic equation `ax^2+bx+c=0` are `x=(-b pm sqrt(b^2-4ac))/(2a)`.

Whether these roots are real or complex numbers, when you average the two, the square roots cancel and you get `x=((-2b)/(2a))/2=-b/(2a)`.

This clearly gives the vertex when there are `x`-intercepts (since the vertex is (horizontally) halfway between the intercepts). It also happens to work when the roots are complex numbers.