Quadratics

Quadratic expressions can be handled quickly by rewriting them in *vertex form*: $a(x-h{)}^2+k$. This reveals the parabola’s turning point $(h,k)$ and makes it easy to read off the minimum or maximum value. To get vertex form, complete the square: factor out the leading coefficient $a$ if it isn’t 1, then add and subtract $(b/2a{)}^2$ inside the brackets. The resulting $h$ equals $-\frac{b}{2a}$ and $k$ is the value of the original expression at $x=h$. Knowing the vertex also gives the discriminant $\Delta =b^2-4ac$, which tells whether the quadratic has real roots. With these tools you can find minima, solve equations, and locate vertices in just a few steps.