O-Level Surds: worked solution

Question

Let p = √5 + 2. Rewrite the quotient (p + 2)/(p - 1) so that it appears as a√5 + b with a and b rational. [3]

Step 1: Substitute p = √5 + 2 into the top and the bottom separately. Numerator: p + 2 = (√5 + 2) + 2 = √5 + 4. Denominator: p - 1 = (√5 + 2) - 1 = √5 + 1. So the quotient is (√5 + 4)/(√5 + 1). Step 2: The denominator still contains a surd, so rationalise it. Multiply the top and the bottom by the conjugate of √5 + 1, which is √5 - 1. This does not change the value, because (√5 - 1)/(√5 - 1) = 1. (√5 + 4)/(√5 + 1) × (√5 - 1)/(√5 - 1) Step 3: Work out the new denominator using the difference of two squares, (a + b)(a - b) = a² - b². (√5 + 1)(√5 - 1) = (√5)² - 1² = 5 - 1 = 4. The surd has gone, which was the point of the conjugate. Step 4: Expand the new numerator term by term. (√5 + 4)(√5 - 1) = √5 × √5 + √5 × (-1) + 4 × √5 + 4 × (-1) = 5 - √5 + 4√5 - 4 = (5 - 4) + (-1 + 4)√5 = 1 + 3√5. Step 5: Put the two parts together. (1 + 3√5)/4. Step 6: Split the single fraction into two so it matches the required form a√5 + b. (3√5)/4 + 1/4 = (3/4)√5 + 1/4. So a = 3/4 and b = 1/4, and both are rational as required.