If the equation 4^x - a 2^ x+1 + a^2 - 5a + 14 = 0 has two distinct real roots whose sum is 3 , then the value of a is :

Step-by-step solution

The given equation is 4^x - a 2^ x+1 + a^2 - 5a + 14 = 0 . Let t = 2^x . The equation transforms into a quadratic in t : t^2 - 2at + a^2 - 5a + 14 = 0 Let the two distinct real roots of the original equation be x₁ and x₂ . We are given that x₁ + x₂ = 3 . The corresponding roots of the quadratic in t are t₁ = 2^ x₁ and t₂ = 2^ x₂ . The product of these roots is: t₁ t₂ = 2^ x₁ 2^ x₂ = 2^ x₁ + x₂ = 2^3 = 8 From Vieta's formulas for the quadratic in t , the product of the roots is also given by the constant term: t₁ t₂