0580/43 Oct/Nov 2022, Question 2: The Quadratic Formula and Completing the Square

Part (ai) · 1 mark

Simplify the product of two powers of p by adding indices.

1. 1p³ × p¹¹ = p³⁺¹¹ = p¹⁴

Answerp¹⁴

Part (aii) · 2 marks

Simplify a quotient of terms in m by dividing coefficients and subtracting indices.

1. 118 ÷ 3 = 6, m⁶ ÷ m² = m⁴
2. 2= 6m⁴

Answer6m⁴

Part (aiii) · 3 marks

Simplify a fractional power expression raised to −1/3, applying index laws to 27, 64, x⁹ and y²⁷.

1. 1= (64/(27x⁹y²⁷))^(1/3)
2. 2∛64 = 4, ∛27 = 3, ∛(x⁹) = x³, ∛(y²⁷) = y⁹
3. 3= 4/(3x³y⁹)

Answer4/(3x³y⁹) or (4x⁻³y⁻⁹)/3

Part (b) · 2 marks

Use the nth term 3n² to write the first three terms of the sequence.

1. 13 × 1² = 3, 3 × 2² = 12, 3 × 3² = 27
2. 23, 12, 27

Answer3, 12, 27

Part (ci) · 2 marks

Find the nth term of the linear sequence 13, 16, 19, 22, 25.

1. 1Common difference 3, so 3n + k
2. 2First term: 3 × 1 + k = 13, k = 10
3. 3nth term = 3n + 10

Answer3n + 10

Part (cii) · 2 marks

Find the nth term of the sequence 3, 17, 55, 129, 251.

1. 1Differences 14, 38, 74, 122 → 24, 36, 48 → 12, 12: cubic with 12 ÷ 6 = 2, so 2n³
2. 2Sequence − 2n³: 3 − 2 = 1, 17 − 16 = 1, 55 − 54 = 1
3. 3nth term = 2n³ + 1

Answer2n³ + 1

Part (d) · 3 marks

Solve the equation (3x − 22)/4 = 23 for x.

1. 13x − 22 = 4 × 23 = 92
2. 23x = 114
3. 3x = 38

Answer38

Part (e) · 4 marks

Use the quadratic formula on 3x² + 8x − 20 = 0, showing working and rounding answers to 2 decimal places.

1. 1x = (−8 ± √(8² − 4 × 3 × (−20)))/(2 × 3)
2. 2= (−8 ± √304)/6
3. 3x = (−8 + 17.435…)/6 = 1.572… or x = (−8 − 17.435…)/6 = −4.239…
4. 4x = −4.24, 1.57

Answer−4.24, 1.57

More past-paper questions on the quadratic formula and completing the square

• 0580/22 May/Jun 2026 Q21 · 3 marks
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