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4PM1/01R Jan 2020, Question 2: The Quadratic Formula and Completing the Square

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 5 marks

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The question, in short

For f(x) = 7 + 4x − x², rewrite in completed-square form a − b(x + c)² with integers a, b, c, then use it to find the maximum value and where it occurs.

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Part (a) · 3 marks

Express f(x) = 7 + 4x − x² as a − b(x + c)², finding integer values of a, b and c.

  1. 17 + 4x − x² = 7 − (x² − 4x)
  2. 2x² − 4x = (x − 2)² − 4
  3. 3f(x) = 7 − (x − 2)² + 4 = 11 − (x − 2)²

Answer11 − (x − 2)² (a = 11, b = 1, c = −2)

Part (b) · 2 marks

Using your completed-square form, state the maximum value of f(x) and the x value at which it occurs.

  1. 1(x − 2)² ≥ 0, so f(x) ≤ 11
  2. 2Maximum value 11 when x − 2 = 0, i.e. x = 2

Answer(i) 11 (ii) x = 2

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