Sheet № 21 · Higher only · AQA · Edexcel · OCR
Completing the Square –
Completing the square is a technique that rewrites a quadratic expression into a form that reveals the turning point of its graph. It is one of three methods for solving quadratic equations — alongside factorising and the quadratic formula — and it is the only one that directly shows you the minimum or maximum point of a parabola. This to
§Key definitions
Question:
Write x² − 8x + 5 in the form (x + p)² + q and state the turning point of y = x² − 8x + 5.
Answer:
(x − 4)² − 11; turning point is (4, −11).
Check:
x = −3 + √10 ≈ 0.162. Substituting: (0.162)² + 6(0.162) − 1 ≈ 0.026 + 0.974 − 1 = 0 ✓
Q1:
Write x² + 10x + 18 in the form (x + p)² + q.
Q2:
Find the turning point of y = x² − 4x + 9.
§Formulas to memorise
x² + bx + c = (x + b/2)² − (b/2)² + c
For a monic quadratic x² + bx + c, halve the coefficient of x, square it, then adjust the constant.
The turning point of the parabola y = ax² + bx + c is at (−p, q).
Start with — x² + bx + c.
Halve the coefficient of x. — If b = 6, then half is 3.
Write the perfect square bracket: — (x + 3)².
Expanding (x + 3)² gives x² + 6x + 9. — But you only had x² + 6x (without the +9), so you have introduced an extra 9.
Subtract the extra: — (x + 3)² − 9.
Add back the original constant c: — (x + 3)² − 9 + c.
Factor out a — from the x² and x terms: a(x² + (b/a)x) + c.
Worked example
Write x² − 8x + 5 in the form (x + p)² + q and state the turning point of y = x² − 8x + 5.
Working:
⚠ Common mistakes
- ✗Forgetting to subtract the square you added. When you write (x + 3)², you have introduced +9. You must subtract 9 to keep the expression equivalent.
- ✗Getting the sign inside the bracket wrong. If the coefficient of x is −8, the bracket should contain −4, not +4. Halve including the sign.
- ✗Not factoring out a for non-monic quadratics. If the coefficient of x² is not 1, you must factor it out before completing the square.
- ✗Losing the ± when solving. Taking a square root produces two values. Writing only the positive root loses one solution.
- ✗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 in the bracket.
✦ Exam tips
- →Know what the question is asking. "Write in the form (x + p)² + q" means complete the square. "Solve by completing the square" means you must also set = 0 and find x.
- →Use completing the square to find turning points. If a question asks for the minimum value of a quadratic or the coordinates of the vertex, this is the method to use.
- →The answer to "find the minimum value" is q, not the full coordinate. Read carefully whether they want the minimum value (just q) or the turning point (both coordinates).
- →Practise non-monic examples. These are rarer but high-value questions. Being comfortable with 2(x + p)² + q format can earn you full marks where many students drop out.