Concept preview

What you will be able to do

1. 01Turn $x^2 + bx$ into a perfect square by adding $(b/2)^2$.
2. 02Rewrite any quadratic in vertex form $a(x - h)^2 + k$ by completing the square.
3. 03Read the vertex, max/min, and range from the completed form.
4. 04Solve a quadratic by completing the square, isolating the square and taking $\pm\sqrt{\ }$.
5. 05Handle a leading coefficient $\ne 1$ by factoring it out first.

Vertex form let you read the vertex at a glance, and the quadratic formula solved anything you threw at it — but have you wondered where those two came from? Both are built from a single, elegant technique called completing the square. The idea is to rewrite a quadratic so the variable $x$ appears inside just one perfect square, plus a constant. Once the messy spread-out $x$-terms are folded into a single square, you can read the vertex straight off or undo the square to solve. Master this one move and the rest of the chapter falls into place.