Let f x be a quadratic polynomial with leading coefficient 1 such that f 0 = p , p ≠ 0 , and f 1 = 1 3 . If the equations f x = 0 and f o f o f o f x = 0 have a common real root, then f - 3 is equal to ______.

Step-by-step solution

Let f x = x - α x - β It is given that f 0 = p ⇒ α β = p and f 1 = 1 3    ⇒   1 - α 1 - β = 1 3 Now let us assume that α is the common root of f x = 0 and f o f o f o f   x = 0 f o f o f o f   x = 0 ⇒ f o f o f o f   α = 0 ⇒   f o f o f   0 = 0 ⇒   f o f p = 0 So, f p is either α or β . Now assuming p - α p - β = α ⇒ α β - α α β - β = α