O-Level Equations and inequalities: worked solution

Question

Solve 2x² − x − 252 = 0, giving the roots correct to three decimal places.

The equation is 2x² − x − 252 = 0, so comparing with ax² + bx + c = 0: a = 2, b = −1, c = −252. First work out the discriminant b² − 4ac to see whether it factorises neatly: b² − 4ac = (−1)² − 4(2)(−252) = 1 + 2016 = 2017. 2017 is not a perfect square (44² = 1936 and 45² = 2025), so the roots are irrational and we must use the quadratic formula. x = (−b ± √(b² − 4ac)) / (2a) x = (1 ± √2017) / 4. Evaluate the surd: √2017 = 44.91102... (since 44.91² = 2016.908 and 44.92² = 2017.806). First root (take +): x = (1 + 44.91102)/4 = 45.91102/4 = 11.477756... Second root (take −): x = (1 − 44.91102)/4 = −43.91102/4 = −10.977756... Rounding each to three decimal places (look at the 4th decimal, a 7 in both cases, so round up): x ≈ 11.478 or x ≈ −10.978.