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0580/22 Oct/Nov 2025, Question 20: The Quadratic Formula and Completing the Square

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 11 marks

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The question, in short

Given f(x) = 21/(2x − 1), x ≠ 1/2, and g(x) = 3x + 4, evaluate g(2), the composite gf(−1), find the inverse f⁻¹(x), and solve f(x) = g(x).

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Part (ai) · 1 mark

Calculate the value of g(2).

  1. 1g(2) = 3 × 2 + 4 = 10

Answer10

Part (aii) · 2 marks

Work out gf(−1), applying f first then g.

  1. 1f(−1) = 21/(2 × (−1) − 1) = 21/(−3) = −7
  2. 2gf(−1) = g(−7) = 3 × (−7) + 4 = −17

Answer−17

Part (aiii) · 3 marks

Find an expression for the inverse function f⁻¹(x).

  1. 1y = 21/(2x − 1), swap: x = 21/(2y − 1)
  2. 22y − 1 = 21/x
  3. 32y = 21/x + 1 = (21 + x)/x
  4. 4f⁻¹(x) = (21 + x)/(2x)

Answer(21 + x)/(2x) or 21/(2x) + 1/2

Part (b) · 5 marks

Solve the equation f(x) = g(x) for x.

  1. 121/(2x − 1) = 3x + 4
  2. 221 = (2x − 1)(3x + 4) = 6x² − 3x + 8x − 4
  3. 36x² + 5x − 25 = 0
  4. 4(3x − 5)(2x + 5) = 0
  5. 5x = 5/3 or x = −5/2

Answer5/3 and −5/2

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Also searched as: 0580/22/O/N/25 · 0580_w25_qp_22 · 0580 w25 22 · question 20.
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