O-Level Equations and inequalities: worked solution

Question

(iii) Solve 10x² − 95x − 21 = 0, giving each root to 2 decimal places.

The question asks for roots to 2 decimal places, which signals that the quadratic does not factorise (no pair of whole numbers multiplies to 10 × (−21) = −210 and adds to −95), so use the quadratic formula. x = (−b ± √(b² − 4ac))/(2a) with a = 10, b = −95, c = −21 Work out the discriminant first, taking care with the two negative signs: b² − 4ac = (−95)² − 4 × 10 × (−21) (−95)² = 9025 −4 × 10 × (−21) = +840 b² − 4ac = 9025 + 840 = 9865 √9865 = 99.3227... It is positive, so there are two real roots. Now substitute. Note that −b = −(−95) = +95 and 2a = 20: x = (95 ± 99.3227)/20 First root: x = (95 + 99.3227)/20 = 194.3227/20 = 9.7161... Second root: x = (95 − 99.3227)/20 = −4.3227/20 = −0.2161... Round each to 2 decimal places: x = 9.72 or x = −0.22