O-Level Surds: worked solution

Question

Without a calculator, determine the integers a and b for which x√18 − √72 = x√3 has solution (a + 2√b)/5.

Method: simplify each surd, gather the x terms, make x the subject, rationalise the denominator with the conjugate, then compare with the given form to read off a and b. Step 1 - simplify the surds by taking out the largest square factor. √18 = √(9 × 2) = 3√2 √72 = √(36 × 2) = 6√2 The equation x√18 − √72 = x√3 becomes 3√2 x − 6√2 = √3 x Step 2 - put both x terms on the same side. 3√2 x − √3 x = 6√2 Step 3 - factorise x out. x(3√2 − √3) = 6√2 x = 6√2 / (3√2 − √3) Step 4 - rationalise the denominator using the conjugate 3√2 + √3. Multiply top and bottom by 3√2 + √3. Denominator: (3√2 − √3)(3√2 + √3) = (3√2)² − (√3)² = 18 − 3 = 15 Numerator: 6√2(3√2 + √3) = 6√2 × 3√2 + 6√2 × √3 6√2 × 3√2 = 18 × 2 = 36 6√2 × √3 = 6√6 so the numerator is 36 + 6√6 x = (36 + 6√6)/15 Step 5 - reduce the fraction so it matches the required form. 36, 6 and 15 all divide by 3: x = (12 + 2√6)/5 Step 6 - compare with (a + 2√b)/5. (12 + 2√6)/5 against (a + 2√b)/5 gives a = 12 and b = 6.