# √(x + 8) + √(x + 15) = √(9x + 40)

$$\sqrt{(x+8)}+\sqrt{(x+15)}-\sqrt{(9x+40)}$$

2021-06-03

$$\sqrt{(x+8)}+\sqrt{(x+15)}-\sqrt{(9x+40)}$$
$$\displaystyle{x}+{8}+{2}\sqrt{(x+8)(x+15)}+{x}+{15}={9}{x}+{40}$$
$$2\sqrt{(x+8)(x+15)}=7x+17$$
$$\displaystyle{4}{\left({x}+{8}\right)}{\left({x}+{15}\right)}={49}{x}^{{2}}+{238}{x}+{289}$$
$$\displaystyle{4}{x}^{{2}}+{92}{x}+{480}={49}{x}^{{2}}+{238}{x}+{283}$$
$$\displaystyle{45}{x}^{{2}}+{146}{x}-{191}={0}$$
$$\displaystyle{\left({45}{x}+{191}\right)}{\left({x}-{1}\right)}={0}$$
$$\displaystyle{x}=-{\left(\frac{{191}}{{45}}\right)}{\quad\text{or}\quad}{x}={1}$$
Note at this point that the couple of times we squared both sides could have introduced extraneous solutions. So we'll need to test each individually. x=1x=1 is easy to check and turns out to be a solution. For −191/45, I'd recommend putting it in WolframAlpha. Doing so, you should see that
$$\sqrt{{\left(-{\left(\frac{{191}}{{45}}\right)}+{8}\right)}}+\sqrt{{\left(-{\left(\frac{{191}}{{45}}\right)}+{15}\right)}}\sim{5.22}$$
$$\sqrt{{\left({9}{\left(-{\left(\frac{{191}}{{45}}\right)}\right)}+{40}\right)}}\sim{1.34}$$
So x=1 is the only solution.