حل السؤال 21، 4MA1/1HR يونيو 2022: الجذور الصماء

1. 1Expand the denominator: (√2 − 1)² = (√2 − 1)(√2 − 1) = 2 − √2 − √2 + 1 = 3 − 2√2
2. 2√8 = √4 × √2 = 2√2, so the numerator is 3 + 2√2
3. 3The expression is (3 + 2√2) ÷ (3 − 2√2)
4. 4Multiply numerator and denominator by 3 + 2√2:
5. 5Numerator: (3 + 2√2)(3 + 2√2) = 9 + 6√2 + 6√2 + 8 = 17 + 12√2
6. 6Denominator: (3 − 2√2)(3 + 2√2) = 9 + 6√2 − 6√2 − 8 = 1
7. 7So the expression = (17 + 12√2) ÷ 1 = 17 + 12√2
8. 812√2 = √(12² × 2) = √288
9. 9The expression is 17 + √288, so p = 17 and q = 288

Answer17 + √288