If the sum of all distinct real roots of the equation (4^x - a)(8^x - 7 4^x + 14 2^x - 8) = 0 is 6 , then the value of the real parameter a is

Step-by-step solution

Let y = 2^x . Then 4^x = y^2 and 8^x = y^3 . The given equation can be written as: (y^2 - a)(y^3 - 7y^2 + 14y - 8) = 0 Consider the cubic polynomial y^3 - 7y^2 + 14y - 8 = 0 . By inspection, y = 1 is a root since 1 - 7 + 14 - 8 = 0 . Factoring out (y - 1) , we get: (y - 1)(y^2 - 6y + 8) = 0 (y - 1)(y - 2)(y - 4) = 0 This gives y = 1, 2, 4 . Since y = 2^x , the corresponding values of x are: 2^x = 1 x = 0 2^x = 2 x = 1 2^x = 4 x = 2 The sum of these distinct real roots is 0 + 1 + 2 = 3 . For the total sum of all dis