Completing the Square & the Quadratic Formula — AQA GCSE Mathematics

SPEC A18 (HIGHER) • ALGEBRA

Completing the Square & Quadratic Formula

⭐ High-priority. Higher tier only. When factorising fails, these two methods always work — and they unlock turning points, proof questions, and solutions to any degree of accuracy.

Why These Two Methods?

Not every quadratic factorises nicely. Completing the square and the quadratic formula always work — and each unlocks something factorising can't:

What each method adds

Completing the square: reveals the turning point directly, proves positivity, gives exact roots in surd form, and is the basis for deriving the quadratic formula.

Quadratic formula: delivers decimal solutions to any precision, handles non-factorable quadratics, and lets the discriminant reveal the number of real roots.

Authentic exam evidence

Jun 2023 P1H Q21: "Write x² − 6x − 15 in the form (x − a)² − b" — 2 marks.

Nov 2024 P1H Q18: (a) Express x² + 8x − 5 as (x + a)² − b; (b) Turning point of y = (x − 7)² + 8.

Jun 2022 P2H Q29: "y = x² − 18x + 70. By completing the square, find the turning point" — 3 marks.

Nov 2023 P2H Q18: "By completing the square, prove x² + 6x + 13 is always positive" — 3 marks.

Key Terms

Completed square form

A quadratic rewritten as a ( x − h )² + k. In this form, the turning point is at (h, k), and the sign of a says whether it's a minimum (a > 0) or maximum (a < 0).

Quadratic formula

For ax² + bx + c = 0, the solutions are x = (− b ± √( b ² − 4 ac )) / (2 a ). The formula is on the formula sheet at Higher tier, but it's quicker if you know it.

Discriminant

The expression b ² − 4 ac (the bit under the square root). Its sign tells you the number of real roots: positive → 2, zero → 1 (repeated), negative → 0.

Completing the Square — Monic Case

For x² + bx + c (coefficient 1 on x²): halve b , square it, add and subtract.

x² + bx + c = (x + b/2)² − (b/2)² + c

Why it works

The "perfect square" (x + b/2)² expands to x² + bx + (b/2)². So it contains b/2 too many constants — subtract (b/2)² to cancel, then keep the original c.

Sign of a

For x² − 6x + …, the completed form is (x− 3)² + …, not (x + 3)². Jun 2023 P1H Q21 mark scheme explicitly says "(x + 3)² scores B0" — wrong sign kills both marks.

Monic Worked Example

Ex

Write x² − 6x − 15 in the form (x − a)² − b

Checking (and scoring)

Expand to verify: (x − 3)² − 24 = x² − 6x + 9 − 24 = x² − 6x − 15 ✓. Mark scheme: B2 for (x−3)² − 24; B1 for (x−3)² alone, or a = 3 implied. Writing (x+3)² scores B0.

Non-Monic Completing the Square

When a ≠ 1, factor a out of the first two terms, complete the square inside, then distribute.

Remember to multiply the constant out

When you distribute the 2 across [(x+2)² − 4], you must multiply both parts: 2 × (x+2)² and 2 × (−4). Forgetting the second gives "(x+2)² − 4 − 3 = (x+2)² − 7" which is wrong.

Finding Turning Points

The completed square form a ( x − h )² + k tells you the turning point directly: it's at ( h , k ).

The parabola y = x² − 6x + 5 can be rewritten as (x − 3)² − 4. The completed-square form tells you the turning point is at (3, −4) — a minimum because the coefficient of x² is positive.

Sign flip on h

For (x− 3)², the x-coordinate of the TP is +3 (not −3). Remember: (x − h)² has TP at x = +h. For (x + 3)² the TP would be at x = −3.

Non-monic case works the same way

From slide 6: 2x² + 8x − 3 = 2(x + 2)² − 11. Reading the completed form: h = −2 (sign flip from (x + 2)²), k = −11. So the turning point is at (−2, −11). Minimum because a = 2 > 0.

Proving Always Positive (or Negative)

A common Higher question: "By completing the square, prove that … is always positive." The reasoning uses the fact that any square is ≥ 0.

Ex

Nov 2023 P2H Q18 authentic — 3 marks

By completing the square, prove that x² + 6x + 13 is always positive.

The critical phrase

You must explicitly state that "(anything)² is always ≥ 0" or equivalent. Leaving the logical step implicit loses the final mark. Mark scheme expects: completed form + the ≥ 0 justification + final conclusion.

The Quadratic Formula

For any quadratic ax² + bx + c = 0, the solutions are given by:

x = (−b ± √(b² − 4ac)) / 2a

When to use it

When the question specifies a decimal accuracy (e.g. "to 2 d.p.").

When factorising doesn't work or isn't obvious.

When the question explicitly says "use the quadratic formula".

Calculator bracket discipline

Type the whole numerator inside brackets: (-b + √(b²-4ac)) ÷ (2a). Missing the numerator brackets gives -b + √(...) ÷ 2a — which only divides the last term. This is the #1 calculator error.

Sign of c

When c is negative, the "−4ac" becomes positive. Example: c = −4, a = 2 → −4ac = −4 × 2 × (−4) = +32. Always substitute with brackets: 4 × 2 × (−4) = −32, so b² − 4ac = b² + 32.

The Discriminant b ² − 4 ac

The expression under the square root is called the discriminant. Its sign tells you how many real roots the quadratic has — no need to solve.

DiscriminantReal rootsParabola vs x-axis

b ² − 4 ac > 02 distinct real rootsCrosses the x-axis twice

b ² − 4 ac = 01 repeated real rootTouches the x-axis once (vertex on axis)

b ² − 4 ac < 0No real rootsNever crosses the x-axis

Worked example — show no real solutions

Show that 3x² + 2x + 5 = 0 has no real solutions. a = 3, b = 2, c = 5

b² − 4ac = 2² − 4(3)(5) = 4 − 60 = −56

−56 < 0, so no real solutions. ✓

Worked example — show two distinct real roots

Show that 2x² − 5x + 3 = 0 has two distinct real roots. a = 2, b = −5, c = 3

b² − 4ac = (−5)² − 4(2)(3) = 25 − 24 = 1

1 > 0, so two distinct real roots. ✓

MCE Chain for Completing the Square

Every completing-the-square question splits marks across these stages.

MMethod. Identify what's asked: form conversion, turning point, prove positivity, or solve. Decide whether the quadratic is monic or non-monic. Halve b.

CCalculate. Write (x ± b/2)², subtract (b/2)², add constant. For non-monic, factor a out first. Apply the result to answer the specific question.

EEvaluate. Expand your completed form to verify it equals the original. For turning points, read h (with sign flipped) and k directly.

Worked Example: Setup

Ex

Higher — 3 marks (Jun 2022 P2H Q29 authentic)

The equation of a curve is y = x ² − 18 x + 70.

By completing the square, work out the coordinates of the turning point.

M — Method

Monic quadratic. Halve the coefficient of x: −18 ÷ 2 = −9. Write (x − 9)² (note the negative sign). Subtract (−9)² = 81. Add the original constant 70.

Then read the turning point from the completed square form.

Worked Example: Calculate & Evaluate

C — Calculate

y = x² − 18x + 70

(x − 9)² = x² − 18x + 81

y = (x − 9)² − 81 + 70

= (x − 9)² − 11

Turning point of a(x − h)² + k is at (h, k).

Here h = 9, k = −11.

E — Evaluate

Verify: (x − 9)² − 11

= x² − 18x + 81 − 11

= x² − 18x + 70 ✓

Turning point: (9, −11)This is a minimum since the coefficient of x² is positive (parabola opens upward).

Mark breakdown (Jun 2022 mark scheme)

M1 for (x − 9)² (or a = 9). M1dep for (x − 9)² − 11 (the completed form). A1 for (9, −11). Note: SC1 is awarded if you state (9, −11) correctly without showing the completing-the-square working — the question says "you must show your working", so the MS penalises method-less answers with only 1 mark.

Weak vs Strong: Sign of c in the Discriminant

When c is negative, the "−4ac" becomes positive. Getting this sign wrong is the single biggest quadratic-formula error.

Weak student

Solves 2x² + 5x − 4 = 0 by computing:

Missed the minus sign on c. Ends up with a negative discriminant and claims "no real roots" — wrong.

Strong student

Substitutes c = −4 in brackets:

Positive discriminant → two real roots. Correctly proceeds to x ≈ 0.64 or −3.14.

Substitute with brackets always

b² − 4ac should be written as b² − 4 × a × (c), with the value of c in brackets if it's negative. Two negatives = positive.

Examiner Alerts

Wrong sign on a

For x² − 6x + …, you need (x− 3)², not (x + 3)². Jun 2023 P1H Q21 MS: "(x + 3)² B0". Halving b keeps its sign.

Forgetting to subtract ( b /2)²

Nov 2024 P1H Q18(a): most students got (x + 4)² correctly but then wrote "(x + 4)² − 5" — forgetting to subtract 4² = 16. The correct answer is (x + 4)² − 21.

Sign of c in discriminant

When c is negative, −4ac is positive. Substitute with brackets and carry through both minus signs carefully.

Rounding too early

Keep √(discriminant) to at least 4 d.p. until the final step. Rounding √57 to 7.55 instead of 7.5498 gives answers 0.001 outside the mark-scheme range — lost A marks.

"You must show your working"

When the question specifies you must show the completing-the-square method (e.g. Jun22 P2H Q29), getting the final answer without visible completing-the-square working earns only 1 SC mark out of 3.

Practice Question 1

Complete the square (Jun 2023 P1H Q21 authentic). Write x² − 6x − 15 in the form (x − a)² − b, where a and b are integers. [2 marks]

M

Halve the coefficient of x: −6 ÷ 2 = −3. Start with (x − 3)².

C

(x − 3)² = x² − 6x + 9. Subtract the extra 9, keep the original −15: (x − 3)² − 9 − 15 = (x − 3)² − 24.

E

a = 3, b = 24. Final form: ( x − 3)² − 24. Mark scheme: B2 full; B1 partial ((x−3)² alone or a=3). Answer (x+3)² scores B0 (wrong sign).

Practice Question 2

Complete the square + turning point (Nov 2024 P1H Q18 authentic).

(a) Express x² + 8x − 5 in the form (x + a)² − b, where a and b are integers. [2 marks]

(b) A curve has the equation y = (x − 7)² + 8. Write down the coordinates of the turning point. [2 marks]

(a) C

Halve 8 → 4. (x + 4)² = x² + 8x + 16. Subtract 16, keep −5: (x + 4)² − 16 − 5 = ( x + 4)² − 21. So a = 4, b = 21. [B2]

(b)

The form is a(x − h)² + k with a = 1, h = 7, k = 8. Turning point: (7, 8). Minimum since a > 0. [B2; B1 each correct coordinate]

Practice Question 3

Quadratic formula. Solve 2x² + 5x − 4 = 0, giving your answers to 2 decimal places. [3 marks]

M

Identify a = 2, b = 5, c = −4. Substitute into x = (−b ± √(b² − 4ac)) / 2a. Use brackets for the c = −4 substitution.

C

b² − 4ac = 25 − 4(2)(−4) = 25 + 32 = 57. √57 ≈ 7.5498…

x = (−5 ± 7.5498) / 4.

x₁ = 2.5498/4 ≈ 0.6375.

x₂ = −12.5498/4 ≈ −3.1375.

E

Rounded to 2 d.p.: x ≈ 0.64 or x ≈ −3.14. Typical mark scheme: M1 substitution in brackets; M1 correct discriminant (57) and √57; A1 both answers to 2 d.p.

Summary

Completing the Square

x² + bx + c = (x + b/2)² − (b/2)² + c

For non-monic: factor a out of first two terms first, then complete the inner square.

Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

Use brackets when substituting values — especially if c is negative.

Discriminant b² − 4ac

> 0: two distinct real roots

= 0: one repeated real root

< 0: no real roots

Top Exam Tip

For ( x − h )² + k, the turning point is at ( h , k ) — note the sign flip on h. Always verify your completed form by expanding. Always keep 4 d.p. of accuracy until the final step. When c is negative, use brackets so you don't lose the sign.