O-Level Surds

Side = (a + b√2), so area = (a + b√2)² = a² + 2b² + 2ab√2. Set equal to 17 - 8√2 and compare rational and irrational parts: a² + 2b² = 17 2ab = -8 => ab = -4 => a = -4/b. Substitute into a² + 2b² = 17: 16/b² + 2b² = 17. Multiply through by b²: 16 + 2b⁴ = 17b² 2b⁴ - 17b² + 16 = 0. (shown)

Method: get the square root by itself on one side, square both sides to remove it, solve the quadratic that results, then test both roots back in the original equation, because squaring can create false solutions. Step 1 - isolate the surd. 8 − √(11 − 2x) = −x Add √(11 − 2x) to both sides: 8 = −x + √(11 − 2x) Add x to both sides: 8 + x = √(11 − 2x) Step 2 - square both sides. (8 + x)² = 11 − 2x Expand the left side: (8 + x)² = 64 + 16x + x² 64 + 16x + x² = 11 − 2x Step 3 - collect everything on one side. x² + 16x + 2x + 64 − 11 = 0 x² + 18x + 53 = 0 Step 4 - solve with the quadratic formula, a = 1, b = 18, c = 53. Discriminant = 18² − 4(1)(53) = 324 − 212 = 112 √112 = √(16 × 7) = 4√7 x = (−18 ± 4√7)/2 = −9 ± 2√7 Step 5 - test both roots, since squaring may have introduced a false one. From Step 1, 8 + x equals a square root, and a square root is never negative, so we need 8 + x ≥ 0, i.e. x ≥ −8. For x = −9 − 2√7 ≈ −9 − 5.29 = −14.29: this is less than −8, so it fails and is rejected. For x = −9 + 2√7 ≈ −9 + 5.29 = −3.71: this is greater than −8, so it is allowed. Check it in the original: 11 − 2(−3.71) = 18.42, √18.42 = 4.29, and 8 − 4.29 = 3.71 = −x. Correct. So x = −9 + 2√7, which is of the form a + b√c with a = −9, b = 2, c = 7.

Method: simplify each surd first, gather every x term on one side, factorise x out, then rationalise the denominator. Step 1 - simplify the surds by pulling out the largest square factor. √18 = √(9 × 2) = √9 × √2 = 3√2 √32 = √(16 × 2) = √16 × √2 = 4√2 The equation x√18 = 3x + √32 becomes 3√2 x = 3x + 4√2 Step 2 - collect the x terms on the left. 3√2 x − 3x = 4√2 Step 3 - factorise x out of the left side. x(3√2 − 3) = 4√2 The bracket has a common factor of 3: 3x(√2 − 1) = 4√2 Step 4 - make x the subject. x = 4√2 / [3(√2 − 1)] Step 5 - rationalise the denominator by multiplying top and bottom by the conjugate √2 + 1. The conjugate works because (√2 − 1)(√2 + 1) = (√2)² − 1² = 2 − 1 = 1. x = [4√2 × (√2 + 1)] / [3 × (√2 − 1)(√2 + 1)] = [4√2 × √2 + 4√2 × 1] / (3 × 1) = (4 × 2 + 4√2)/3 = (8 + 4√2)/3 This matches the required form (c + d√2)/3 with c = 8 and d = 4.