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4PM1/01 Jun 2024, Question 2: The Quadratic Formula and Completing the Square

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

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The question, in short

For f(x) = 2x² + 4x + 9, express it as A(x + B)² + C with integer A, B, C, then use this to find where 1/f(x) is greatest and that greatest value.

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

Part (a) · 3 marks

Find the integer values of A, B and C when f(x) = 2x² + 4x + 9 is written as A(x + B)² + C.

  1. 12x² + 4x + 9 = 2(x² + 2x) + 9
  2. 2= 2[(x + 1)² − 1] + 9
  3. 3= 2(x + 1)² + 7
  4. 4A = 2, B = 1, C = 7

AnswerA = 2, B = 1, C = 7

Part (b) · 2 marks

Using your completed-square form, find the x giving the maximum of 1/f(x), and state that maximum value.

  1. 11/f(x) is a maximum when f(x) is a minimum
  2. 2f(x) = 2(x + 1)² + 7 is least when x = −1
  3. 3Minimum of f(x) is 7
  4. 4Maximum of 1/f(x) = 1/7

Answer(i) x = −1 (ii) 1/7

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