Completing the Square

Completing the square turns a quadratic with an x² term and a linear x term into a square of a binomial, which is often easier to solve or interpret. It works because if you expand (x + b/2)², you get x² + bx + b²/4, so subtracting b²/4 keeps the expression equivalent to the original. This rearrangement exposes the vertex form of a parabola and makes many algebraic steps simpler.

Use this when solving quadratic equations, especially if factoring is awkward or impossible. It is also useful in graphing parabolas, finding turning points, and converting quadratic expressions into a form that is easier to integrate or analyze.

• Forgetting to subtract the square of half the coefficient, which changes the value of the expression
• Using b instead of b/2 inside the bracket, which does not form a perfect square
• Dropping the x² term or writing the result as a solved equation instead of an equivalent expression