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4PM1/02 Jun 2022, Question 5: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 9 marks

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The question, in short

Quadratic 2x² + (6 + 2p)x + 2p = 0 has roots α and β; express sum and product in p, verify a relation, then use α − β = 3 to find p.

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Part (a) · 2 marks

Write α + β and αβ in terms of p using the coefficients of the quadratic.

  1. 1α + β = −b/a = −(6 + 2p)/2 = −3 − p
  2. 2αβ = c/a = 2p/2 = p

Answerα + β = −3 − p; αβ = p

Part (b) · 4 marks

Show that (α − β)² equals 9 + 2p + p².

  1. 1(α − β)² = α² + β² − 2αβ
  2. 2= (α + β)² − 4αβ
  3. 3= (−3 − p)² − 4p = 9 + 6p + p² − 4p
  4. 4= 9 + 2p + p²

Answer(α − β)² = 9 + 2p + p² shown

Part (c) · 3 marks

Given α − β = 3, determine the possible values of p.

  1. 1(α − β)² = 9
  2. 29 + 2p + p² = 9 ⇒ p² + 2p = 0
  3. 3p(p + 2) = 0 ⇒ p = 0 or p = −2

Answerp = 0 or p = −2

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