Completing the Square

Completing the square means adding the square of half the x-coefficient to both sides, so the quadratic side becomes a perfect square trinomial and the square root property finishes.

Solve x² + 6x = 7. Half of 6 is 3; its square is 9. Add 9 to both sides: x² + 6x + 9 = 16. The left side is now the perfect square (x + 3)², so (x + 3)² = 16, and the square root property gives x + 3 = ±4. Two endings: x = 1 or x = −7. Check both: 1 + 6 = 7 and 49 − 42 = 7. The method manufactures the pattern it needs — the added 9 is exactly what makes the left side a square — and it works on every quadratic, which is why the next formula can be derived from it.