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4PM1/01R Jun 2024, Question 10: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 12 marks

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The question, in short

Quadratic 2x² + kx + 4 = 0 has roots α, β with k < 0, α > β, and α² − β² = 7√17/4. Show k = −7, then build a quadratic with roots α − β and α + β.

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

Part (a) · 8 marks

Using root sums, products and the given difference of squares, show that k = −7.

  1. 1α + β = −k/2, αβ = 2
  2. 2(α − β)² = (α + β)² − 4αβ
  3. 3α² − β² = (α + β)(α − β) = (−k/2)√(k²/4 − 8) = 7√17/4
  4. 4Square: (k²/4)(k²/4 − 8) = 833/16, so k⁴ − 32k² − 833 = 0
  5. 5(k² − 49)(k² + 17) = 0
  6. 6k² = 49 and k < 0 gives k = −7

Answerk = −7 shown

Part (b) · 4 marks

Hence form a quadratic equation whose roots are α − β and α + β.

  1. 1α + β = 7/2 and α − β = √17/2
  2. 2Product = α² − β² = 7√17/4
  3. 3Sum = √17/2 + 7/2 = (√17 + 7)/2
  4. 4x² − ((√17 + 7)/2)x + 7√17/4 = 0
  5. 54x² − 2(√17 + 7)x + 7√17 = 0

Answer4x² − 2(√17 + 7)x + 7√17 = 0

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