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4PM1/02 Jan 2022, Question 7: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Given f(x) = 2x² + px + q with integer p, q and p + q = −4, p > 0, and g(x) = 14x² + 37x + 14 whose roots are α/β and β/α, find p and q, then show α² − β² = −3√65/4.

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

Part (a) · 9 marks

Using root relationships for f and g without solving g(x) = 0, find the integer values of p and q given p + q = −4 and p > 0.

  1. 1α + β = −p/2, αβ = q/2
  2. 2α/β + β/α = −37/14
  3. 3α/β + β/α = ((α + β)² − 2αβ)/(αβ) = (p²/4 − q)/(q/2) = (p² − 4q)/(2q)
  4. 4Use q = −p − 4: (p² + 4p + 16)/(2(−p − 4)) = −37/14
  5. 57p² − 9p − 36 = 0
  6. 6(7p + 12)(p − 3) = 0, p > 0 so p = 3
  7. 7q = −4 − 3 = −7

Answerp = 3, q = −7

Part (b) · 4 marks

Given α > β, show that α² − β² equals −3√65/4.

  1. 1α² − β² = (α − β)(α + β)
  2. 2(α − β)² = (α + β)² − 4αβ = 9/4 + 14 = 65/4
  3. 3α > β so α − β = √65/2
  4. 4α² − β² = (√65/2)(−3/2) = −3√65/4

Answerα² − β² = −3√65/4 shown

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