XoaX.net's MathML: Completing the Square

The minimum or maximum value of a parabola can be found by completing the square, as shown. The squared term is greater than or equal to zero. So, it is minimized when it is zero, at the double root. The size of the multiplier a determines whether it is a minimum or maximum. Finally, the minimum or maximum is given by the value of the last term.

\begin{matrix} y & = & a x^{2} + b x + c \\ = & a (x^{2} + \frac{b}{a} x) + c \\ = & a (x^{2} + \frac{b}{a} x + {(\frac{b}{2 a})}^{2} - {(\frac{b}{2 a})}^{2}) + c \\ = & a ({(x + \frac{b}{2 a})}^{2} - {(\frac{b}{2 a})}^{2}) + c \\ = & a {(x + \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a} + c \\ = & a {(x + \frac{b}{2 a})}^{2} + \frac{4 a c - b^{2}}{4 a} \end{matrix}

To make the situation clearer, we can write $y = a z^{2} + d$ , where $z = x + \frac{b}{2 a}$ and $d = \frac{4 a c - b^{2}}{4 a}$ .