O-Level Equations and inequalities: worked solution

Question

Solve x^2 - 25x - 50000 = 0, giving each answer correct to two decimal places.

The answers are wanted to two decimal places, which tells you the roots are not whole numbers, so factorising will not work here. Use the quadratic formula. For ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / (2a). Comparing x² - 25x - 50000 = 0 with ax² + bx + c = 0: a = 1, b = -25, c = -50000. Work out the discriminant first: b² - 4ac = (-25)² - 4 × 1 × (-50000) = 625 + 200000 = 200625. Take the square root: √200625 = 447.9118... (200625 = 625 × 321, so this is exactly 25√321). Now substitute into the formula, remembering that -b = -(-25) = 25: x = (25 ± 447.9118...) / 2. Taking the + sign: x = (25 + 447.9118...)/2 = 472.9118.../2 = 236.4559... Taking the - sign: x = (25 - 447.9118...)/2 = -422.9118.../2 = -211.4559... Round each to two decimal places (the third decimal is 5 or more in each case, so the second decimal rounds up in size): x = 236.46 or x = -211.46.