Completing the square
[[completing-the-square|Completing the square]] turns any quadratic into a perfect-square trinomial plus a leftover number, so you can finish with the square-root property. The trick rests on the pattern (x + p)^2 = x^2 + 2px + p^2. To rebuild that pattern, take half of the x-coefficient and square it — that is the number you add (and subtract) to keep the equation balanced.
Solve x^2 + 6x - 7 = 0 by completing the square.
x^2 + 6x = 7 (move constant right)
half of 6 is 3, and 3^2 = 9:
x^2 + 6x + 9 = 7 + 9 (add 9 to BOTH sides)
(x + 3)^2 = 16 (left side is a perfect square)
x + 3 = ±4 (square-root property)
x = -3 ± 4
x = 1 or x = -7The quadratic formula: completing the square, once and for all
Completing the square works on every quadratic — so what if you do it once on the general form a x^2 + b x + c = 0, with letters instead of numbers? The answer that falls out is the [[quadratic-formula|quadratic formula]], a single expression that solves any quadratic the moment you read off a, b, c. It is not a separate trick; it is completing the square, frozen into a formula.
Quadratic formula:
-b ± sqrt(b^2 - 4ac)
x = --------------------
2a
Solve 2x^2 + 3x - 5 = 0 (a = 2, b = 3, c = -5):
x = ( -3 ± sqrt(3^2 - 4·2·(-5)) ) / (2·2)
x = ( -3 ± sqrt(9 + 40) ) / 4
x = ( -3 ± sqrt(49) ) / 4
x = ( -3 ± 7 ) / 4
x = 4/4 = 1 or x = -10/4 = -5/2