Definition: Completing the square rewrites a quadratic expression as a squared binomial plus a constant by adding the number that makes x^2 + bx a perfect square trinomial.
What is the idea?
The expression x2 + bx becomes a perfect square when you add (b/2)2. Then it equals (x + b/2)2. To keep the equation balanced, add the same amount to both sides, or add and subtract it.
How does it solve an equation?
Move the constant right, add (b/2)2 to both sides, factor the left side as a square, and take square roots.
Example. Solve x2 + 10x – 3 = 0. Write x2 + 10x = 3. Half of 10 is 5, and 5 squared is 25. So x2 + 10x + 25 = 28, which gives (x + 5)2 = 28. Then x + 5 = ± the square root of 28, so x = -5 ± 2 times the square root of 7, about 0.29 or -10.29.
Where do students slip?
They add the new number to one side only. When the leading coefficient is not 1, divide through first. Also, half of b must be squared, not just halved. The method also reveals the vertex form of a parabola.