Let f(x) be a quadratic polynomial such that f(-1) + f(3) = 0 and f(2) = 0 . If the equations f(x) = 0 and x^2 + px + 2p = 0 have at least one common root, then the sum of all possible real values of p is equal to

Step-by-step solution

Let f(x) = ax^2 + bx + c . Given f(2) = 0 : 4a + 2b + c = 0 (i) Given f(-1) + f(3) = 0 : (a - b + c) + (9a + 3b + c) = 0 10a + 2b + 2c = 0 5a + b + c = 0 (ii) Subtracting equation (i) from equation (ii) gives: (5a + b + c) - (4a + 2b + c) = 0 a - b = 0 a = b Substituting b = a into equation (ii) : 5a + a + c = 0 c = -6a Thus, the polynomial is f(x) = ax^2 + ax - 6a = a(x^2 + x - 6) . The roots of f(x) = 0 are given by x^2 + x - 6 = 0 (x + 3)(x - 2) = 0 So, the roots are x = -3 and x = 2 . The equation x^2 + px + 2p