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

Nov 3, 2015

Average the roots that come from the quadratic formula to see that the $x$-coordinate of the vertex of $a {x}^{2} + b x + c = 0$ is $x = - \frac{b}{2 a}$.
The quadratic formula says the roots of the quadratic equation $a {x}^{2} + b x + c = 0$ are $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.
Whether these roots are real or complex numbers, when you average the two, the square roots cancel and you get $x = \frac{\frac{- 2 b}{2 a}}{2} = - \frac{b}{2 a}$.
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.