O-Level Equations and inequalities: worked solution

Question

Solve 6v² − 3v − 193 = 0, giving your answers to 3 significant figures.

The equation 6v² − 3v − 193 = 0 is a quadratic in v. Comparing with av² + bv + c = 0: a = 6, b = −3, c = −193. The answers are asked for to 3 significant figures, which is a signal that the roots are not whole numbers, so the quadratic will not factorise. Use the quadratic formula v = (−b ± √(b² − 4ac)) / (2a). Step 1: work out the discriminant b² − 4ac. b² = (−3)² = 9. 4ac = 4 × 6 × (−193) = −4632, and subtracting a negative is the same as adding, so b² − 4ac = 9 + 4632 = 4641. Step 2: substitute into the formula. Note −b = −(−3) = 3 and 2a = 12. v = (3 ± √4641) / 12. Step 3: evaluate. √4641 = 68.1249... (it is not a whole number, since 68² = 4624 and 69² = 4761). Taking the plus sign: v = (3 + 68.1249) / 12 = 71.1249 / 12 = 5.92707... Taking the minus sign: v = (3 − 68.1249) / 12 = −65.1249 / 12 = −5.42707... Step 4: round each root to 3 significant figures. 5.92707... rounds to 5.93 (the fourth figure is 7, so round up). −5.42707... rounds to −5.43 (the fourth figure is 7, so round up in size). v = 5.93 or v = −5.43.