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4MA1/1HR Jun 2024, Question 20: 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 2x² − 11x + 9 in completed-square form a(x − b)² − c, then use it to locate the minimum point P of the curve y = 2(x − 3)² − 11(x − 3) + 9.

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

Part (a) · 3 marks

Find the numbers a, b and c so that 2x² − 11x + 9 equals a(x − b)² − c.

  1. 1Take out the 2: 2x² − 11x + 9 = 2(x² − 11/2 x) + 9
  2. 2Complete the square: x² − 11/2 x = (x − 11/4)² − 121/16
  3. 3So 2(x − 11/4)² − 121/8 + 9, and −121/8 + 72/8 = −49/8
  4. 42(x − 11/4)² − 49/8

Answer2(x − 11/4)² − 49/8

Part (b) · 2 marks

Find the coordinates of P, the minimum point on curve C.

  1. 1Put X = x − 3; by part (a), 2X² − 11X + 9 = 2(X − 11/4)² − 49/8
  2. 2So y = 2(x − 3 − 11/4)² − 49/8 = 2(x − 23/4)² − 49/8
  3. 3The squared term is never negative, so y is least when x = 23/4
  4. 4P = (23/4, −49/8)

Answer(23/4, −49/8)

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