arenceabigns

2021-09-15

$\sqrt{\left(x+8\right)}+\sqrt{\left(x+15\right)}=\sqrt{\left(9x+40\right)}$

coffentw

$\sqrt{\left(x+8\right)}+\sqrt{\left(x+15\right)}=\sqrt{\left(9x+40\right)}$
$x+8+2\sqrt{\left(\left(x+8\right)\left(x+15\right)\right)}+x+15=9x+40$
$2\surd \left(\left(x+8\right)\left(x+15\right)=7x+17$
$4\left(x+8\right)\left(x+15\right)=49{x}^{2}+238x+289$
$4{x}^{2}+92x+480=49{x}^{2}+238x+283$
$45{x}^{2}+146x-191=0$
$\left(45x+191\right)\left(x-1\right)=0$
$x=-\left(\frac{191}{45}\right)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=1$
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.

Jeffrey Jordon

Answer is given below (on video)

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