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MathidiaMathematics Academy
Quadratic equations and graphs
Lesson 17 of 75
Easy practice

Completing the square

Objective. Rewrite quadratics to expose turning points and roots.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Completing the square is developed here as connected GCSE mathematics, not a memorised shortcut. Connect factorisation, roots, turning points and graphical solutions. The lesson moves from a precise representation and key definitions to fluent procedure, linked reasoning and an unfamiliar application. Learners compare possible methods, state restrictions and assumptions, preserve exact values where appropriate, and justify accuracy, notation and units before evaluating whether the result is mathematically and contextually reasonable.

Worked example

What does the discriminant determine?

the number of real roots
Illustrated card showing the number of real roots
  1. 1Translate the information into a labelled diagram, table, graph or algebraic statement and identify every constraint.
  2. 2Select the most efficient GCSE method, explain why its conditions apply, and keep exact values until approximation is requested.
  3. 3Carry out each transformation on a separate justified line, tracking signs, units, domains and required accuracy.
  4. 4Verify with substitution, an inverse method, estimation or a second representation. The checked result is the number of real roots.

Answer: the number of real roots

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. A common error in completing the square is applying a familiar rule before checking its conditions, domain or direction. Name the structure first, retain exact values, and test the final result against every original constraint.

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