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.
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Here are the solutions for 5a.i and 5a.ii. 5a.i. Solve f(x) = g(-(1)/(2)) Step 1: Identify the functions f(x) and g(x). We are given f(x) = x-2 and g(x) = 2x^2 - 1. Step 2: Evaluate g(-(1)/(2)). Substitute x = -(1)/(2) into the expression for g(x): 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) Step 3: Set f(x) equal to the value found for g(-(1)/(2)). f(x) = -(1)/(2) x-2 = -(1)/(2) Step 4: Solve for x. Add 2 to both sides of the equation: 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 Step 1: Substitute the expressions for f(x) and g(x) into the equation. (x-2) + (2x^2 - 1) = 0 Step 2: Simplify and rearrange the equation into standard quadratic form ax^2 + bx + c = 0. 2x^2 + x - 2 - 1 = 0 2x^2 + x - 3 = 0 Step 3: Factor the quadratic equation. We look for two numbers that multiply to 2 × (-3) = -6 and add to 1. These numbers are 3 and -2. 2x^2 + 3x - 2x - 3 = 0 Factor by grouping: x(2x+3) - 1(2x+3) = 0 (x-1)(2x+3) = 0 Step 4: Set each factor to zero and solve for x. x-1 = 0 x = 1 2x+3 = 0 2x = -3 x = -(3)/(2) The solutions are x = 1 or x = -(3)/(2). That's 2 down. 3 left today — send the next one.