Sheet № 90 · Higher only · AQA · Edexcel · OCR
Completing the Square Method –
Completing the square is a Higher tier technique that rewrites a quadratic expression into a form that reveals the turning point of the parabola. It is also a powerful method for solving quadratic equations, especially when the question demands exact (surd) answers.
§Key definitions
Question:
Write x² + 6x + 2 in the form (x + p)² + q.
Answer:
(x + 3)² - 7
Q1 (Higher):
Write x² + 8x + 10 in the form (x + p)² + q.
Q2 (Higher):
Find the minimum value of x² - 12x + 40.
Q3 (Higher):
Solve x² - 2x - 5 = 0 by completing the square, giving answers in surd form.
§Formulas to memorise
x² + bx + c = (x + b/2)² - (b/2)² + c
Turning point of y = (x + p)² + q is (-p, q)
So x² + 4x - 3 = (x + 2)² - 4 - 3 = (x + 2)² - 7.
Worked example
Write x² + 6x + 2 in the form (x + p)² + q.
Working:
⚠ Common mistakes
- ✗Forgetting to subtract the square you introduced. When you write (x + 3)², you create an extra +9. You must subtract 9 to keep the expression equivalent to the original.
- ✗Getting the sign of p wrong. If the coefficient of x is -8, then p = -4, not +4. Always halve including the sign.
- ✗Confusing the turning point coordinates. In (x + 3)² - 7, the turning point is (-3, -7), not (3, -7). The x-coordinate is the opposite sign of what appears inside the bracket.
✦ Exam tips
- →Read whether the question asks you to "write in the form (x + p)² + q" or to "solve by completing the square." The first requires only the rewrite; the second requires finding x values.
- →For "find the minimum value" questions, complete the square and state q. The minimum value is q, not the full coordinate pair.
- →Practise with negative coefficients of x, as these are where most students make sign errors.