Answer
The solutions are $5+\sqrt{17}$ and $5-\sqrt{17}$.
Work Step by Step
$ x^{2}-10x+8=0\qquad$ ... write left side in the form $x^{2}+bx$ (add $-8$ to each side).
$ x^{2}-10x=-8\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{-10}{2})^{2}=5^{2}=25\qquad$ ...add $25$ to each side of the expression
$ x^{2}-10x+25=-8+25\qquad$ ... write left side as a binomial squared.
$(x-5)^{2}=-8+25\qquad$ ...simplify.
$(x-5)^{2}=17\qquad$ ...take square roots of each side.
$ x-5=\pm\sqrt{17}\qquad$ ...add $5$ to each side
$ x-5+5=\pm\sqrt{17}+5\qquad$ ...simplify.
$x=5\pm\sqrt{17}$
