Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p , where p 0 . If the equations f(x) = 0 and f(f(f(x))) = 0 have a common real root, and f(p) + f(2) = 6 , then the value of f(4) is ______.

Step-by-step solution

Let be the common real root of f(x) = 0 and f(f(f(x))) = 0 . Since f( ) = 0 , we have: f(f(f( ))) = f(f(0)) = f(p) = 0 . We are given that f(p) + f(2) = 6 . Substituting f(p) = 0 , we get 0 + f(2) = 6 f(2) = 6 . Since f(p) = 0 , p is a root of the quadratic equation f(x) = 0 . Let the roots of f(x) = 0 be p and q . Since f(x) is a monic quadratic polynomial, the product of its roots is equal to the constant term f(0) . Thus, p q = f(0) = p . Since p 0 , we can divide by p to get q = 1 . Therefore, the roots of f(x)