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: Identify the given roots. Let the roots be = (2)/(3) and = -1. Step 2: Calculate the sum of the roots. + = (2)/(3) + (-1) = (2)/(3) - (3)/(3) = -(1)/(3) Step 3: Calculate the product of the roots. = (2)/(3) × (-1) = -(2)/(3) Step 4: Form the quadratic equation using the formula x^2 - ( + )x + = 0. Substitute the sum and product of the roots into the formula: x^2 - (-(1)/(3))x + (-(2)/(3)) = 0 x^2 + (1)/(3)x - (2)/(3) = 0 Step 5: Multiply the entire equation by 3 to eliminate the fractions. 3 (x^2 + (1)/(3)x - (2)/(3)) = 3 × 0 3x^2 + x - 2 = 0 Comparing this with the given options: A. 3x^2 - x - 2 = 0 B. 3x^2 + x + 2 = 0 C. 3x^2 + x - 2 = 0 D. 3x^2 + x - 1 = 0 The derived equation matches option C. The final answer is C. 3x^2 + x - 2 = 0.