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WMA11/01 Jun 2021, Question 2: The Quadratic Formula and Completing the Square

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Given cubic f(x) = ax³ + (6a + 8)x² − a²x with positive a and f(−1) = 32, show a = 3, solve f(x) = 0, then solve two related equations in y and z.

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

Part (ai) · 3 marks

Substitute x = −1 into f, set equal to 32, and show the only positive solution is a = 3.

  1. 1f(−1) = −a + (6a + 8) + a² = 32
  2. 2a² + 5a − 24 = 0
  3. 3(a + 8)(a − 3) = 0
  4. 4a = 3 or a = −8; a is positive, so a = 3

AnswerShown: a² + 5a − 24 = 0 ⇒ (a + 8)(a − 3) = 0 ⇒ a = 3 (a = −8 rejected as a > 0)

Part (aii) · 2 marks

With a = 3, solve the cubic f(x) = 0 for x.

  1. 1f(x) = 3x³ + 26x² − 9x = 0
  2. 2x(3x² + 26x − 9) = 0
  3. 3x(3x − 1)(x + 9) = 0
  4. 4x = 0, x = 1/3, x = −9

Answerx = 0, 1/3, −9

Part (bi) · 3 marks

Using the cubic result, find all real solutions of 3y + 26y^(2/3) − 9y^(1/3) = 0.

  1. 1With x = y^(1/3), the equation is 3x³ + 26x² − 9x = 0
  2. 2So y^(1/3) = 0, 1/3 or −9
  3. 3y = 0³ = 0, y = (1/3)³ = 1/27, y = (−9)³ = −729

Answery = 0, 1/27, −729

Part (bii) · 2 marks

Using the cubic result, find all real solutions of 3(9^(3z)) + 26(9^(2z)) − 9(9^z) = 0.

  1. 1With x = 9^z, the equation is 3x³ + 26x² − 9x = 0, so 9^z = 0, 1/3 or −9
  2. 29^z is always positive, so only 9^z = 1/3 is possible
  3. 39^z = 3^(2z) = 3⁻¹, so 2z = −1
  4. 4z = −1/2

Answerz = −1/2

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