Completing the square
Rewrite ax² + bx + c as a(x + p)² + q. Used to find min/max and to derive the quadratic formula.
Higher-tier technique. The minimum (or maximum if a is negative) of the quadratic occurs at x = −p, with value q. Useful for sketching and for solving quadratics that don't factorise.
Worked examples
x² + 6x + 5 = (x + 3)² − 4.
x² − 4x + 7 = (x − 2)² + 3. Min value 3 at x = 2.
2x² + 8x + 1 = 2(x + 2)² − 7. Factor out leading coefficient first.
Frequently asked questions
How does it derive the formula?
Complete the square on ax² + bx + c = 0; the rearrangement gives the quadratic formula.
Why find minimum?
Useful for word problems (e.g. minimum cost) and graph sketching (vertex form).