Completing the square

Some quadratics won't factor nicely — x² + 6x + 2 = 0 has no integer pair with sum 6 and product 2. The rescue is a technique with an unusually honest name: we are literally going to complete a square.

The picture first

Draw x² as a square and bx as a rectangle. Cut the rectangle into two strips of width b/2 and wrap them around two sides of the square. The result is almost a bigger square of side x + b/2 — it's missing exactly one corner piece of area (b/2)². So:

$x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2$

Add the missing corner and the expression becomes a perfect square — the pattern from the special-products lesson, now used as a tool.

Solving with it

Solve x² + 6x + 2 = 0. Here b = 6, so the magic corner is (6/2)² = 9.

Two exact solutions, from a trinomial that refused to factor. Note the ±: both √7 and −√7 square to 7, and forgetting the negative branch silently discards half the answers.

Vertex form, for free

The same manipulation rewrites y = x² + 6x + 2 as y = (x + 3)² − 7. That's vertex form: the parabola's lowest point sits at (−3, −7), readable directly. One technique, two payoffs — exact solutions and the graph's turning point.

Why this matters

Completing the square is the rare method that always works — no luck with factor pairs required. Run it one time on the fully general equation ax² + bx + c = 0 and out comes a formula you can reuse forever. That's the next lesson, and it's the finale of the course.