This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions for 5a.i and 5a.ii, presented without explicit step labels. 5a.i. Solve f(x) = g(-(1)/(2)) Given f(x) = x-2 and g(x) = 2x^2 - 1. First, evaluate g(-(1)/(2)): g(-(1)/(2)) = 2(-(1)/(2))^2 - 1 g(-(1)/(2)) = 2((1)/(4)) - 1 g(-(1)/(2)) = (1)/(2) - 1 g(-(1)/(2)) = -(1)/(2) Now, set f(x) equal to this value: f(x) = -(1)/(2) x-2 = -(1)/(2) x = 2 - (1)/(2) x = (4)/(2) - (1)/(2) x = (3)/(2) The solution is x = (3)/(2). 5a.ii. Solve f(x) + g(x) = 0 Substitute the expressions for f(x) and g(x): (x-2) + (2x^2 - 1) = 0 2x^2 + x - 3 = 0 Factor the quadratic equation: 2x^2 + 3x - 2x - 3 = 0 x(2x+3) - 1(2x+3) = 0 (x-1)(2x+3) = 0 Set each factor to zero to find the solutions for x: x-1 = 0 x = 1 2x+3 = 0 2x = -3 x = -(3)/(2) The solutions are x = 1 or x = -(3)/(2). 3 done, 2 left today. You're making progress.