IGCSE Maths / Quadratic equations, curves and inequalities
Solving quadratics by completing the square
Edexcel International GCSE Maths (4MA1), chapter 22: Quadratic equations, curves and inequalities, specification point 2.7b. Free lesson.
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Completing the square is a method for solving quadratic equations, including ones that do not factorise. It turns the equation into a squared bracket, which you can undo by taking both square roots.
How completing the square works
The method rewrites the quadratic expression as a squared bracket plus or minus a number. To form the bracket, halve the coefficient of x. Squaring the bracket adds an extra number, so subtract that same number to keep the expression unchanged. This method works even when a quadratic does not factorise. Factorising with whole numbers only works for some quadratics. A quadratic that does not factorise can still have solutions.
Solve x² − 4x − 3 = 0
Halving −4 gives −2, so the squared bracket is (x − 2)². Squaring the bracket adds 4, so subtract 4 to keep the expression equal to the original. Then move the numbers to the right.
Why both square roots are needed
When you take the square root of both sides, write ± straight away. This is because √7 and −√7 both square to 7: squaring removes the minus sign. A quadratic has two solutions here, so keeping only the positive root leaves out one answer. For x² − 4x − 3 = 0, the answers are 2 + √7 and 2 − √7. To 2 decimal places, these are 4.65 and −0.65.
Solve x² + 6x + 4 = 0
Halving 6 gives 3, and squaring 3 gives 9. Subtract 9 to complete the square, then rearrange and take both square roots. Keep the answers in surd form, which means leaving the square root exact.