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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 rewrites a quadratic in the form a(x + p)² + q, revealing the turning point directly at (−p, q). For x² + bx + c, halve the coefficient of x to get p = b/2, then write (x + p)² − p² + c, since expanding (x + p)² gives x² + bx + p². For example, x² + 6x + 1 becomes (x + 3)² − 9 + 1 = (x + 3)² − 8. This form shows the minimum value of the expression is q when a is positive, occurring at x = −p.

Worked example

Write x² − 8x + 5 in completed square form a(x + p)² + q, identifying the turning point.

Diagram of a square
  1. 1Halve the coefficient of x: −8 ÷ 2 = −4.
  2. 2Form the bracket: (x − 4)².
  3. 3Expand (x − 4)² gives x² − 8x + 16, so subtract 16 and add the original 5: (x − 4)² − 16 + 5.
  4. 4Simplify: (x − 4)² − 11.

Answer: (x − 4)² − 11

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Common mistake. Pupils forget to subtract p² after forming the squared bracket, leaving an expression that is not equivalent to the original, such as writing x² + 6x + 1 as (x + 3)² + 1. Always subtract p² to compensate.

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