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0606/22 Feb/Mar 2026, Question 1: The Quadratic Formula and Completing the Square

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 8 marks

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The question, in short

For f(x) = 2x² − 6x + 4, complete the square, give its range, find the largest k making g one-to-one on x ≤ k, and form gh(x) with h(x) = e^(3x) − 1.

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

Part (a) · 3 marks

Rewrite f(x) = 2x² − 6x + 4 as a(x + b)² + c, stating constants a, b and c.

  1. 12x² − 6x + 4 = 2(x² − 3x) + 4
  2. 2= 2[(x − 3/2)² − 9/4] + 4
  3. 3= 2(x − 3/2)² − 1/2

Answer2(x − 3/2)² − 1/2

Part (b) · 1 mark

Using your completed-square form, state the range of f.

  1. 1Minimum value of 2(x − 3/2)² − 1/2 is −1/2
  2. 2Range f(x) ≥ −1/2

Answerf(x) ≥ −1/2

Part (c) · 1 mark

For g(x) = 2x² − 6x + 4 with x ≤ k, give the largest k so g has an inverse.

  1. 1The vertex is at x = 3/2
  2. 2g is one-one for x ≤ 3/2, so the largest k is 3/2

Answer3/2

Part (d) · 3 marks

With h(x) = e^(3x) − 1 for x ≤ 0, find and simplify gh(x), given it exists.

  1. 1gh(x) = 2(e^(3x) − 1)² − 6(e^(3x) − 1) + 4
  2. 2= 2e^(6x) − 4e^(3x) + 2 − 6e^(3x) + 6 + 4
  3. 3= 2e^(6x) − 10e^(3x) + 12

Answergh(x) = 2e^(6x) − 10e^(3x) + 12

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Also searched as: 0606/22/F/M/26 · 0606_m26_qp_22 · 0606 m26 22 · question 1.
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