Completing the Square

Algebra 1 · Axiom Academy

Before any formula, you can literally build a perfect square out of tiles — and see exactly which piece is missing. The whole idea, in one picture An expression like x² + 6x looks like pure symbols, but it has a shape. A big square of side x covers the x² part; the 6x is six thin strips. Lay them out and they almost — but not quite — fill a bigger square. The whole trick called completing the square is just finding the one piece that finishes it. Watch the strips split evenly along two sides of the big square, opening up an empty corner. Then the exact number of unit tiles needed to fill that corner drops in, and the figure snaps into a perfect square of side x + 3 . Take half of the 6 strips for each side — 3 and 3 — and the corner that's left is exactly 3 by 3. How many tiles finish the corner? Same expression, x² + 6x , already arranged into the L. Drag the slider to drop unit tiles into the empty corner. Too few and there's a gap; too many and they spill past the edge. Find the count that fills it exactly and the whole figure locks into a perfect square. Exactly 9 = 3 × 3 — half of 6, squared. That's the missing corner, and the square is whole. It's always half of b, squared Change the coefficient of x and the picture rebuilds itself. The strips always split into two equal arms of b/2 , so the corner is always a b/2 by b/2 square — that's (b/2)² unit tiles. Drag through the values and watch the corner count follow the rule every single time.

This is the written version of the interactive lesson above. See the full Algebra 1 course.