For the following two (02) items: Let f(x) = ax^2 + bx + c be a quadratic polynomial such that f(1) = f(4) = 2 . Further, 2 is a root of f(x) = 0 . What is the other root of f(x) = 0 ?

Step-by-step solution

Given f(1) = f(4) = 2 , the quadratic polynomial can be written as f(x) = a(x-1)(x-4) + 2 . Since 2 is a root of f(x) = 0 , we have f(2) = 0 . Substituting x = 2 in the expression for f(x) : a(2-1)(2-4) + 2 = 0 a(1)(-2) + 2 = 0 -2a + 2 = 0 a = 1 Thus, the polynomial is f(x) = 1(x-1)(x-4) + 2 = x^2 - 5x + 6 . To find the roots of f(x) = 0 , we solve x^2 - 5x + 6 = 0 . (x-2)(x-3) = 0 The roots are x = 2 and x = 3 . The other root is 3 . Answer: 3