I am trying to solve $400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}} = 0$ for $x$. Thanks to an earlier question I put up here, I know $x = 6- \frac{2}{\sqrt{3}}$, but I want to figure out why. Symbolab gives a ridiculously complicated equation for $x$, and I couldn't figure it out myself when I tried.
Please help me figure out why $x = 6 - \frac{2}{\sqrt{3}}$.
Some of what I tried:
$$400,000-\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}}=0$$
Subtract $400,000$ from both sides.
$$\frac{800,000(6-x)}{\sqrt{4+(6-x)^2}}=400,000$$
Multiply both sides by $\sqrt{4+(6-x)^2}$.
$$800,000(6-x)=400,000\sqrt{4+(6-x)^2}$$
Divide both sides by $800,000$.
$$6-x=\frac{\sqrt{4+(6-x)^2}}{2}$$
Multiply both sides by $2$.
$$12-2x=\sqrt{4+(6-x)^2}$$
Take both sides to the power of $2$.
$$-4x^2-48x+144= -x^2-12x+40$$