Answer
$x=4$ or $x=-2$
$y=1$
Work Step by Step
In order to solve the question, the key step here is to compare and observe the individual parts - imaginary and real parts.
bringing all the terms to one side, we have:
$(x^{2}-2x)-8+y^{2}i-(2y-1)i=0$
since the total sum of the real and imaginary parts are equal to $0$, this implies that the sum of the real parts is equal to $0$ and the sum of the imaginary part is equal to $0$
Therefore,
The real part of the expression $(x^{2}-2x)-8=0$, can be factorized to $(x-4)(x+2)=0$, which gives $x=4$ or $x=-2$
The imaginary part of the expression $y^{2}-(2y-1)=0$, can be simplified to $y^{2}-2y+1=0$. The expression can then be factorized to $(y-1)^{2}=0$, which gives $y=1$