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

Explanation

`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`

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