O-Level Equations and inequalities: worked solution

Question

Find the roots of 4x² + 12x − 495 = 0, each correct to one decimal place.

First check whether 4x² + 12x − 495 factorises with whole numbers. It would need two numbers multiplying to 4 × (−495) = −1980 and adding to 12, and no such pair exists, so use the quadratic formula. For ax² + bx + c = 0 the formula is x = (−b ± √(b² − 4ac))/(2a) Read off the coefficients from 4x² + 12x − 495 = 0, keeping the sign of c: a = 4, b = 12, c = −495 Work out the discriminant b² − 4ac first, taking care with the double negative: b² − 4ac = 12² − 4 × 4 × (−495) = 144 + 7920 = 8064 This is positive, so there are two different real roots. √8064 = 89.7997... (it is not a whole number, so keep the decimals for now) Substitute into the formula, with 2a = 2 × 4 = 8: x = (−12 ± 89.7997...)/8 Take the + sign first: x = (−12 + 89.7997...)/8 = 77.7997.../8 = 9.72497... Now take the − sign: x = (−12 − 89.7997...)/8 = −101.7997.../8 = −12.72497... Round each answer to one decimal place. Look at the second decimal digit to decide. 9.72497... has second decimal 2, so round down: 9.7 −12.72497... has second decimal 2, so round towards 12.7 in size: −12.7 So the roots are x ≈ 9.7 and x ≈ −12.7.