The equation `x^3/4((log)_2x)^(2+(log)_2x-5/4)=sqrt(2)` has at least one real solution exactly three solutions exactly one irrational solution 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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