Let f(x) = x^2 + bx + c be a quadratic polynomial with real coefficients. The equation f(f(x)) = x has four complex roots, two of which are and . If and do not satisfy the equation f(x) = x , and it is given that + = 5 and = 13 , then the value of f(2) is ______.

Step-by-step solution

Consider the equation f(f(x)) - x = 0 . We can rewrite the expression f(f(x)) - x as f(f(x)) - f(x) + f(x) - x . Notice that f(f(x)) - f(x) = (f(x))^2 + b f(x) + c - (x^2 + bx + c) = (f(x))^2 - x^2 + b(f(x) - x) = (f(x) - x)(f(x) + x + b) . Thus, f(f(x)) - x = (f(x) - x)(f(x) + x + b) + (f(x) - x) = (f(x) - x)(f(x) + x + b + 1) . Substitute f(x) = x^2 + bx + c into the second factor: f(x) + x + b + 1 = x^2 + bx + c + x + b + 1 = x^2 + (b+1)x + (b+c+1) . The roots of f(f(x)) = x are the roots of f(x) - x = 0 and x^2