IGCSE Maths / Quadratic equations, curves and inequalities
Completing the square
Edexcel International GCSE Maths (4MA1), chapter 22: Quadratic equations, curves and inequalities, specification point 2.2d. Free lesson.
Read the revision notes · IGCSE Maths past papers
Completing the square rewrites a quadratic in a form that is easier to work with. It helps you keep track of signs and read the turning point from a quadratic curve.
How to complete the square when the coefficient is 1
For x² + bx + c, halve the coefficient of x to get the number in the bracket. Squaring the bracket creates an extra number, so subtract that number back off. This keeps the expression unchanged. The bracket contains half of b, not b itself. Check your result by expanding it back out.
How to handle a coefficient that is not 1
If the coefficient of x² is not 1, first factor it out from the terms containing x. Then complete the square inside the bracket. The constant term that is left outside does not go into the bracket. Only factor from the terms that contain x. After completing the square, multiply everything inside the bracket by the factor outside it.
How to handle a negative coefficient safely
With a negative coefficient, the signs inside the bracket need care. The number subtracted to complete the square is multiplied by the negative factor when you multiply the bracket back out. This can turn a negative into a positive. For −3x² + 12x + 5, factor −3 from the first two terms: −3(x² − 4x) + 5. Inside the bracket, half of −4 is −2. Squaring (x − 2) introduces 4, so subtract 4 inside. When −3 is multiplied back in, −3 × −4 = +12. Then 12 + 5 = 17.
How to read the turning point from the completed square
In the form y = a(x + p)² + q, the squared part is smallest when it equals zero. The turning point is therefore at (−p, q). If a is positive, the curve opens upwards and this point is its lowest point. For y = x² − 6x + 5, completing the square gives y = (x − 3)² − 4. The square is zero when x = 3, so the lowest y-value is −4. The turning point is (3, −4).
A quick check
Use the same halve, square and subtract method when the coefficient of x² is 1. Collect the constants at the end. For x² + 10x + 3, half of 10 is 5.