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4PM1/01 Jun 2021, Question 6: Roots α and β of a Quadratic

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Prove an identity linking (α − β)² to α + β and αβ, then apply it to x² − 7kx + k² = 0 with roots α > β to find α − β and a new quadratic.

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Part (a) · 3 marks

Show that (α − β)² equals (α + β)² − 4αβ.

  1. 1(α − β)² = α² − 2αβ + β²
  2. 2α² + β² = (α + β)² − 2αβ
  3. 3(α − β)² = (α + β)² − 2αβ − 2αβ = (α + β)² − 4αβ

Answer(α − β)² = (α + β)² − 4αβ shown

Part (b) · 3 marks

For x² − 7kx + k² = 0 with roots α and β, α > β, show α − β = 3k√5.

  1. 1α + β = 7k, αβ = k²
  2. 2(α − β)² = (7k)² − 4k² = 45k²
  3. 3α − β = √(45k²) = 3k√5 (positive since α > β, k > 0)

Answerα − β = 3k√5 shown

Part (c) · 4 marks

Hence form a quadratic with roots α + 1 and β − 1, in the form x² + px + q = 0 with p and q in terms of k.

  1. 1Sum = (α + 1) + (β − 1) = α + β = 7k
  2. 2Product = (α + 1)(β − 1) = αβ − (α − β) − 1
  3. 3= k² − 3k√5 − 1
  4. 4x² − 7kx + k² − 3k√5 − 1 = 0

Answerx² − 7kx + k² − 3k√5 − 1 = 0 (p = −7k, q = k² − 3k√5 − 1)

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