The sum of all the real solutions of the equation 25^x - 6 5^ x+1 + 50 = 0 is :

Step-by-step solution

The given equation is 25^x - 6 5^ x+1 + 50 = 0 . This can be rewritten as (5^x)^2 - 30 5^x + 50 = 0 . Let t = 5^x . The equation becomes a quadratic in t : t^2 - 30t + 50 = 0 The discriminant of this quadratic is = (-30)^2 - 4(1)(50) = 900 - 200 = 700 > 0 . Since the sum of the roots is 30 > 0 and the product of the roots is 50 > 0 , both roots t₁ and t₂ are real and positive. Let the corresponding solutions for x be x₁ and x₂ . We have 5^ x₁ = t₁ and 5^ x₂ = t₂ . The product of the roots in t is: t₁ t₂ = 50 Substi