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4MA1/1HR Jun 2022, Question 23: The Quadratic Formula and Completing the Square

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 5 marks

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The question, in short

Rewrite the quadratic 2x² − 12x + 3 as a(x + b)² + c with integer a, b, c, then use that to locate the minimum point M of the curve y = 2(x + 4)² − 12(x + 4) + 3.

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

Part (a) · 3 marks

Put 2x² − 12x + 3 into completed-square form a(x + b)² + c, stating integer values of a, b and c.

  1. 12x² − 12x + 3 = 2(x² − 6x) + 3
  2. 2Complete the square: x² − 6x = (x − 3)² − 9
  3. 3So 2((x − 3)² − 9) + 3 = 2(x − 3)² − 18 + 3
  4. 42(x − 3)² − 15

Answer2(x − 3)² − 15

Part (b) · 2 marks

Determine the coordinates of the minimum point M on the given curve C.

  1. 1Given y = 2(x + 4)² − 12(x + 4) + 3
  2. 2Let u = x + 4, then y = 2u² − 12u + 3
  3. 3From part (a), this is 2(u − 3)² − 15
  4. 4Minimum when u − 3 = 0 → u = 3
  5. 5Then x + 4 = 3 → x = −1
  6. 6y = −15
  7. 7So M is (−1, −15)

Answer(−1, −15)

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