Answer
$x=-5+5\sqrt{3}$
Work Step by Step
Length $\times$ Width $=$ Area of a rectangle.
$ x(x+10)=50\qquad$ ...Distributive property
$ x^{2}+10x=50\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=50+25\qquad$ ...simplify.
$ x^{2}+10x+25=75\qquad$ ...write left side as a binomial squared.
$(x+5)^{2}=75\qquad$ ...take square roots of each side.
$ x+5=\pm\sqrt{75}\qquad$ ...rewrite $\sqrt{75}$ as $\sqrt{25\cdot 3}$
$ x+5=\pm\sqrt{25\cdot 3}\qquad$ ...evaluate $\sqrt{25}$.
$ x+5=\pm 5\sqrt{3}\qquad$ ...add $-5$ to each side.
$ x+5-5=\pm 5\sqrt{3}-5\qquad$ ...simplify.
$x=-5\pm 5\sqrt{3}$
$x=-5+5\sqrt{3}$ or $x=-5-5\sqrt{3}$.
We are calculating length, which cannot be negative. The solution $x=-5-5\sqrt{3}$ is negative, so we discard it.
Solution: $x=-5+5\sqrt{3}$
