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4PM1/01 Nov 2023, Question 2: The Quadratic Formula and Completing the Square

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 7 marks

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The question, in short

For g(x) = 2x² + (1/2)x − 3, complete the square, find its minimum and where it occurs, then use this for h(x) = 2x⁶ + (1/2)x³ − 3.

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

Part (a) · 3 marks

Rewrite g(x) as p(x + q)² + r, giving rational values of p, q and r.

  1. 1g(x) = 2(x² + x/4) − 3
  2. 2= 2[(x + 1/8)² − 1/64] − 3
  3. 3= 2(x + 1/8)² − 1/32 − 3
  4. 4= 2(x + 1/8)² − 97/32

Answerg(x) = 2(x + 1/8)² − 97/32 (p = 2, q = 1/8, r = −97/32)

Part (b) · 2 marks

State the least value of g(x) and the x value giving it.

  1. 1(i) Minimum value = r = −97/32
  2. 2(ii) Minimum when x + 1/8 = 0, x = −1/8

Answer(i) −97/32; (ii) x = −1/8

Part (c) · 2 marks

Using your result, state the least value of h(x) and the x value where it occurs.

  1. 1(i) h(x) = g(x³), so the minimum value is the same: −97/32
  2. 2(ii) x³ = −1/8
  3. 3x = −1/2

Answer(i) −97/32; (ii) x = −1/2

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