lesson
Reveal the turning point
1 · Learn
Completing the square rewrites a quadratic using a squared bracket plus a constant. x² + 6x + 5 = (x + 3)² - 4 because the square introduces 9 and the adjustment removes 4 from it.
A squared real expression is non-negative, so this form identifies a minimum when the leading coefficient is positive. Expand back to check the rearrangement.
2 · Worked example
For y = x² + 6x + 5, the turning point is (-3,-4). The sign inside the bracket must be interpreted carefully.
3 · Your turn
Complete the square for x² - 8x + 10 and give its minimum value.
Check your answer
(x - 4)² - 6; minimum -6 at x = 4.
4 · Apply your learning
Compare the expanded and completed-square forms by substitution and by their graph features.
Adult guidance and safety
This sequence includes extension mathematics and assumes secure earlier work. Check prerequisite understanding and the learner's readiness before assigning extension tasks. Require derivations, accurate diagrams and varied practice; a correct short example is not full assessment evidence. Exam-board and tier-specific pathways are outside this edition.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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