Completing the square

First factor out the coefficient of x² from the quadratic terms. Inside the bracket, halve the coefficient of x and add and subtract its square. Keep any constant outside the bracket until the final simplification.

In a(x − h)² + k, the vertex is (h,k). When a is positive the vertex is a minimum; when a is negative it is a maximum. Reading the sign of h from the bracket is a common source of mistakes: x + 3 means h = −3.

Worked example

Write 2x² + 8x + 5 in square form.

1. Factor: 2(x² + 4x) + 5.
2. Use x² + 4x = (x + 2)² − 4.
3. Simplify 2(x + 2)² − 8 + 5.

Answer: 2(x + 2)² − 3; minimum −3 at x = −2

Revise first: Expanding and factorising.