Find the quadratic equation whose roots are -\(\frac{1}{2}\) and 3

To find the quadratic equation given its roots, we use the fact that for a quadratic equation of the form ax² + bx + c = 0, the roots are given by the formula: x = (-b ± √(b² - 4ac)) / 2a If the roots are given as α and β, then the quadratic equation can be written as: (x - α)(x - β) = 0 Expanding the above equation gives: x² - (α + β)x + αβ = 0 Therefore, to find the quadratic equation whose roots are -\(\frac{1}{2}\) and 3, we substitute α = -\(\frac{1}{2}\) and β = 3 into the equation: x² - (α + β)x + αβ = 0 x² - (-\(\frac{1}{2}\) + 3)x + (-\(\frac{1}{2}\) × 3) = 0 Simplifying the above equation, we get: 2x² - 5x - 3 = 0 Therefore, the quadratic equation whose roots are -\(\frac{1}{2}\) and 3 is 2x² - 5x - 3 = 0. The correct option is (C) 2x² - 5x - 3 = 0.