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Completing the square rewrites a quadratic ax² + bx + c as a(x + p)² + q, revealing the turning point and helping to sketch parabolas, solve equations and find maximum or minimum values. It's a key topic in A-Level and International A-Level (iAL) Pure Mathematics. This free interactive eClassroom course walks through each step with worked examples and instant-feedback questions, so you can practise until it's secure.
How you do it
Halve the coefficient of x, put it in the bracket, then subtract the square of that number. If there is a coefficient in front of x squared, factorise it out first. The completed form gives the turning point straight away.
Before you start
Assumed knowledge: expanding and factorising. Pure 1 is where it is assessed. It underpins turning points and the quadratic formula.
What you learn
Setting the completed form to zero then solves to exact (surd) roots. The course works through turning point, maximum & minimum and building a quadratic from its turning point. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
We teach Completing the Square as one of the Algebra topics in our A-level course library, at the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. Related topics students usually take next include Composite and Inverse Functions, Factor & Remainder Theorem, Functions - Domain and Range, Improper Algebraic Fractions and Indices and Surds. The Algebra revision engine gives you as many extra questions as you want, tracked on your skill tree.
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