4MA1/1HR Jun 2022, Question 21: Surds
Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 4 marks
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The question, in short
Simplify (3 + √8)/(√2 − 1)² and write it as p + √q with p and q integers, showing all stages of working.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Solution · 4 marks
- 1Expand the denominator: (√2 − 1)² = (√2 − 1)(√2 − 1) = 2 − √2 − √2 + 1 = 3 − 2√2
- 2√8 = √4 × √2 = 2√2, so the numerator is 3 + 2√2
- 3The expression is (3 + 2√2) ÷ (3 − 2√2)
- 4Multiply numerator and denominator by 3 + 2√2:
- 5Numerator: (3 + 2√2)(3 + 2√2) = 9 + 6√2 + 6√2 + 8 = 17 + 12√2
- 6Denominator: (3 − 2√2)(3 + 2√2) = 9 + 6√2 − 6√2 − 8 = 1
- 7So the expression = (17 + 12√2) ÷ 1 = 17 + 12√2
- 812√2 = √(12² × 2) = √288
- 9The expression is 17 + √288, so p = 17 and q = 288
Answer17 + √288
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