The equation `x^((3/4)(log_2 x)^2 + (log_2 x) - (5/4))=sqrt(2)`has`(1)`at least one real solution`(2)`exactly three solutions`(3)`exactly one irrational solution`(4)`complex roots

The equation `x^((3/4)(log_2 x)^2 + (log_2 x) - (5/4))=sqrt(2)`has`(1)

1.	The equation `x^((3/4)(log_2 x)^2 + (log_2 x) - (5/4))=sqrt(2)`has`(1)`at least one real solution`(2)`exactly three solutions`(3)`exactly one irrational solution`(4)`complex roots
Answer» `x^((3/4)(log_2x)^2 +log_2x - 5/4) = sqrt2` Taking logs both sides, `=>((3/4)(log_2x)^2 +log_2x - 5/4)log_2 x = log_2 2^(1/2)` `=>((3/4)(log_2x)^2 +log_2x - 5/4)log_2 x = 1/2` Let `log_2x = t` Then, equation becomes, `=>(3/4t^2+t-5/4)t = 1/2` `=>3t^3+4t^2-5t-2 = 0` `=>(t-1)(3t^2+7t+2) = 0` `=>(t-1)(3t^2+6t+t+2) = 0` `=>(t-1)(t+2)(3t+1) = 0` `=>t = 1 or t = -2 or t = -1/3` `=>log_2x = 1 or log_2x = -2 or log_2x = -1/3` `=>x = 2^1,2^(-1/3) , 2^(-2)` So, there are exactly three solutions for `x`.