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
Step 1: Use the given condition f(2) = 0. Substitute x=2 into the function f(x) = x^3 + ax^2 + 5x + c: f(2) = (2)^3 + a(2)^2 + 5(2) + c = 0 8 + 4a + 10 + c = 0 4a + c + 18 = 0 (*) Step 2: Use the given condition f(-1) = 0. Substitute x=-1 into the function f(x) = x^3 + ax^2 + 5x + c: f(-1) = (-1)^3 + a(-1)^2 + 5(-1) + c = 0 -1 + a - 5 + c = 0 a + c - 6 = 0 (**) Step 3: Solve the system of equations for a. From equation (*), we have 4a + c = -18. From equation (**), we have a + c = 6. Subtract equation (*) from equation (): (4a + c) - (a + c) = -18 - 6 4a - a + c - c = -24 3a = -24 a = (-24)/(3) a = -8 The value of a is -8.