O-Level Equations and inequalities: worked solution

Question

Solve 7x² − 350x − 80 000 = 0, giving each solution to 2 decimal places.

Check first whether the expression factorises. The answers are asked for to 2 decimal places, which signals that they are not whole numbers, so factorising will not work. Use the quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)]/(2a) Read off the coefficients from 7x² − 350x − 80 000 = 0: a = 7, b = −350, c = −80 000 Work out the discriminant b² − 4ac first, and take care with the signs: b² = (−350)² = 122 500 4ac = 4 × 7 × (−80 000) = −2 240 000 b² − 4ac = 122 500 − (−2 240 000) = 122 500 + 2 240 000 = 2 362 500 Take the square root: √2 362 500 = 1537.0426... Now substitute into the formula. Note that −b = −(−350) = 350, and 2a = 2 × 7 = 14: x = (350 ± 1537.0426...)/14 Take the plus sign first: x = (350 + 1537.0426...)/14 = 1887.0426.../14 = 134.7887... Now the minus sign: x = (350 − 1537.0426...)/14 = −1187.0426.../14 = −84.7887... Round each to 2 decimal places. In both cases the third decimal is 8, which is 5 or more, so round the second decimal up: x = 134.79 or x = −84.79