Answer
The solutions are $1+2\sqrt{2}$ and $1-2\sqrt{2}$.
Work Step by Step
$ 2n^{2}-4n-14=0\qquad$ ... write left side in the form $x^{2}+bx$ (add $14$ to each side).
$ 2n^{2}-4n=14\qquad$ ...divide each term with $2$.
$ n^{2}-2n=7\qquad$ ...square half the coefficient of $n$.
$(\displaystyle \frac{-2}{2})^{2}=(-1)^{2}=1\qquad$ ...add $1$ to each side of the expression
$ n^{2}-2n+1=7+1\qquad$ ...simplify.
$ n^{2}-2n+1=8\qquad$ ... write left side as a binomial squared.
$(n-1)^{2}=8\qquad$ ...take square roots of each side.
$ n-1=\pm\sqrt{8}\qquad$ ...add $1$ to each side
$ n-1+1=\pm\sqrt{8}+1\qquad$ ...simplify.
$ n=1\pm\sqrt{8}\qquad$ ...rewrite $\sqrt{8}$ as $\sqrt{4\cdot 2.}$
$ n=1\pm\sqrt{4\cdot 2}\qquad$ ...evaluate $\sqrt{4}$.
$n=1\pm 2\sqrt{2}$
