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0606/12 Feb/Mar 2025, Question 5: The Quadratic Formula and Completing the Square

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 8 marks

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The question, in short

Complete the square on 2x² − 2x + 3, then use the restricted domain x ≤ p to find the largest p giving an inverse, state the range, and derive f⁻¹.

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

Part (a) · 3 marks

Express 2x² − 2x + 3 in the completed-square form a(x + b)² + c, stating constants a, b and c.

  1. 12x² − 2x + 3 = 2(x² − x) + 3
  2. 2= 2[(x − 1/2)² − 1/4] + 3
  3. 3= 2(x − 1/2)² + 5/2

Answer2(x − 1/2)² + 5/2

Part (b) · 1 mark

State the largest value of p so that f, for x ≤ p, has an inverse function.

  1. 1The vertex is at x = 1/2
  2. 2f is one-one for x ≤ 1/2, so p = 1/2

Answer1/2

Part (c) · 1 mark

With that p, give the range of f.

  1. 1Minimum value of f is 5/2 at x = 1/2
  2. 2Range f ≥ 5/2

Answerf ≥ 5/2

Part (d) · 3 marks

With that p, obtain a formula for f⁻¹.

  1. 1y = 2(x − 1/2)² + 5/2
  2. 2(x − 1/2)² = (y − 5/2)/2
  3. 3x − 1/2 = −√((y − 5/2)/2) (negative root since x ≤ 1/2)
  4. 4f⁻¹(x) = 1/2 − √((1/2)(x − 5/2))

Answerf⁻¹(x) = 1/2 − √((1/2)(x − 5/2)) or f⁻¹(x) = (2 − √(8x − 20))/4

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