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4MA1/1HR Jan 2023, Question 16: The Quadratic Formula and Completing the Square

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 6 marks

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The question, in short

For f(x) = 2/(3x − 5), x ≠ 5/3, evaluate f at 1/3, derive the inverse function, and rewrite g(x) = 5x² − 20x + 23 in completed-square form a(x − b)² + c.

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

Part (a) · 1 mark

Work out the value of f when x equals 1/3.

  1. 1f(1/3) = 2 / (3 × (1/3) − 5)
  2. 2f(1/3) = 2 / (1 − 5)
  3. 3f(1/3) = 2 / (−4)
  4. 4−0.5

Answer−0.5

Part (b) · 2 marks

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

  1. 1Let y = 2/(3x − 5)
  2. 2Multiply both sides by (3x − 5): y(3x − 5) = 2
  3. 33xy − 5y = 2
  4. 43xy = 2 + 5y
  5. 5Divide both sides by 3y: x = (2 + 5y)/(3y)
  6. 6Write in terms of x: f⁻¹(x) = (2 + 5x)/(3x)

Answer(2 + 5x)/(3x)

Part (c) · 3 marks

Rewrite g(x) as a(x − b)² + c, giving the constants a, b and c.

  1. 1g(x) = 5x² − 20x + 23
  2. 2Factor out 5 from first two terms: 5(x² − 4x) + 23
  3. 3Complete the square: x² − 4x → (x − 2)² − 4
  4. 4So: 5[(x − 2)² − 4] + 23
  5. 5Expand: 5(x − 2)² − 20 + 23
  6. 6Simplify: 5(x − 2)² + 3

Answer5(x − 2)² + 3

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