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0580/23 May/Jun 2026, Question 18: Equations That Become Quadratics

Cambridge IGCSE Mathematics 0580 · Paper 23 (Extended) · Non-calculator · 6 marks

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The question, in short

Rectangle with sides split into 6 cm and x cm, and x cm and 2 cm, with a right-angled triangle of legs x cm cut from a corner; shaded area is 30 cm². Form and solve an equation for x.

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

Part (a) · 3 marks

Using the shaded area of 30 cm², show that the dimensions lead to the equation x² + 16x − 36 = 0.

  1. 1Rectangle area = (x + 6)(x + 2) = x² + 8x + 12
  2. 2Triangle area = ½ × x × x = ½x²
  3. 3x² + 8x + 12 − ½x² = 30
  4. 4½x² + 8x − 18 = 0
  5. 5Multiply by 2: x² + 16x − 36 = 0

Answerx² + 6x + 2x + 12 − ½x² = 30 leading to x² + 16x − 36 = 0

Part (b) · 3 marks

Factorise x² + 16x − 36 = 0 and solve it to find the length x cm.

  1. 1(x + 18)(x − 2) = 0
  2. 2x = −18 or x = 2
  3. 3Length must be positive, so x = 2

Answer2

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Also searched as: 0580/23/M/J/26 · 0580_s26_qp_23 · 0580 s26 23 · question 18.
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