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

1 Answer
Nov 3, 2015

Answer:

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.