Algebra > Quadratics > Completing the Square
Completing the square helps students rewrite quadratics to reveal turning points and solve equations in a different form. The generator now prioritises a standard monic pathway, with selectable worked-example types, structured exam-style questions and optional worked solutions for review.
Key ideas
Students need to understand why (x + a)^2 creates a middle term of 2ax.
The completed square form reveals the vertex of the quadratic graph.
Completing the square can be used to solve quadratic equations.
The method is closely connected to graph transformations.
Non-monic, exact surd and proof-style questions are best treated as extension once the adjustment step is secure.
Common mistakes
Halving the coefficient of x but forgetting to square it.
Writing (x + a)^2 without adjusting the constant term.
Losing negative signs when the completed square form contains subtraction.
Treating the method as a trick rather than an equivalent rewrite.
Teaching notes
Use expansion to justify each completed square identity.
Connect the final form to the minimum or maximum point of the graph.
Start with monic quadratics before moving to harder coefficients.
Check whether students can explain the adjustment to the constant term, not just follow a memorised rule.
Starter ideas
Expand (x + 4)^2.
What number must be added to x^2 + 10x to make a square?
How can x^2 + 6x + 11 be written using (x + 3)^2?
Worksheet generation
The worksheet generator can create support content, prior knowledge, worked examples, fluency practice, problem solving, exam-style questions and answers. This link opens the generator directly on this topic.
Open GCSE worksheet generator