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WMA11/01 Jan 2021, Question 8: The Quadratic Formula and Completing the Square

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

For the curve y = (3x − 2)²(x − 4), find where it is positive, expand it into a cubic polynomial, and find the length PQ where the horizontal line through its y-intercept meets the curve again.

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

Part (a) · 1 mark

State the x-values for which f(x) > 0.

  1. 1(3x − 2)² ≥ 0, and equals 0 only at x = 2/3
  2. 2So f(x) > 0 exactly when x − 4 > 0: x > 4

Answerx > 4

Part (b) · 3 marks

Expand (3x − 2)²(x − 4) into the form ax³ + bx² + cx + d, giving integer coefficients.

  1. 1(3x − 2)² = 9x² − 12x + 4
  2. 2(9x² − 12x + 4)(x − 4) = 9x³ − 36x² − 12x² + 48x + 4x − 16
  3. 3= 9x³ − 48x² + 52x − 16

Answerf(x) = 9x³ − 48x² + 52x − 16

Part (c) · 5 marks

Using algebra, find where the horizontal line through the y-intercept meets the curve again, and give the length PQ as k√3.

  1. 1The y-intercept is −16, so l is y = −16
  2. 29x³ − 48x² + 52x − 16 = −16, so x(9x² − 48x + 52) = 0
  3. 3x = (48 ± √(48² − 4 × 9 × 52))/18 = (48 ± √432)/18
  4. 4x = (16 ± 4√3)/6
  5. 5PQ = (16 + 4√3)/6 − (16 − 4√3)/6 = (4/3)√3

AnswerPQ = (4/3)√3

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