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0580/21 May/Jun 2025, Question 18: Equations That Become Quadratics

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 10 marks

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The question, in short

Anya runs 12 km at x km/h and walks 10 km at (x − 4) km/h; form time expressions, build and solve x² − 2x − 48 = 0, then find her running time.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 1 mark

Give an expression in x for how long the 12 km run takes.

  1. 1Time = distance ÷ speed = 12/x hours

Answer12/x

Part (b) · 1 mark

Give an expression in x for how long the 10 km walk takes.

  1. 1Time = 10/(x − 4) hours

Answer10/(x − 4)

Part (c) · 4 marks

Using the walk lasting 1 hour longer than the run, form an equation in x and show it reduces to x² − 2x − 48 = 0.

  1. 110/(x − 4) = 12/x + 1
  2. 2Multiply by x(x − 4): 10x = 12(x − 4) + x(x − 4)
  3. 310x = 12x − 48 + x² − 4x
  4. 40 = x² − 2x − 48

AnswerLeading to 0 = x² − 2x − 48

Part (d) · 3 marks

Solve x² − 2x − 48 = 0 by factorising.

  1. 1x² − 2x − 48 = 0
  2. 2(x − 8)(x + 6) = 0
  3. 3x = −6 or x = 8

Answer−6, 8

Part (e) · 1 mark

Work out the duration of Anya's run.

  1. 1Speed cannot be negative, so x = 8
  2. 2Time running = 12/8 = 1.5 hours

Answer1.5 or 1 1/2

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