The number of solutions of 16^ ^2 x +16^ ^2 x =10 in 0 x 2 are

Step-by-step solution

The equation 16^ ^2 x + 16^ ^2 x = 10 can be rewritten using the identity ^2 x = 1 - ^2 x to obtain 16^ ^2 x + 16^ 1 - ^2 x = 10 . Let y = ^2 x where 0 y 1 , giving 16^y + 16 16^y = 10 . Substituting t = 16^y with 1 t 16 yields t + 16 t = 10 . Multiplying through by t and rearranging produces the quadratic t^2 - 10t + 16 = 0 , which factors as (t - 2)(t - 8) = 0 . Thus, t = 2 or t = 8 . For 16^y = 2 , write 2^ 4y = 2^1 to obtain y = 1 4 , so ^2 x = 1 4 and x = 1 2 . For 16^y = 8 , write 2^ 4y = 2^3 to obtain y = 3