<s>[INST] $\ \sqrt{x+39}-\sqrt{x+7}=4 $ So I tried to solve this problem for x
$\ \sqrt{x+39}-\sqrt{x+7}=4 $
I multiplied both sides ($\ \sqrt{m}\cdot\sqrt{n}=\sqrt{mn} $)
$\ (\sqrt{x+39}-\sqrt{x+7})^2=16 $
$\ (x+39)-2(x^2+46x+273)-(x+7) $
$\ 0x+32+(-2x^2-92x-546) $
$\ -2x^2-92x-514 $
divide the 2 out
$\ x^2+46x+257=-8 $
$\ x^2+46x+265=0 $
Use the quadratic formula (or scientific calculator) and the answers are -6.752 and -39.248. I know the answer is exactly -3. What went wrong?
 [/INST] You made two mistakes when squaring. The correct equation is
$$(x + 39) - 2\sqrt{(x+39)(x+7)} + (x + 7) = 16$$
This can be rearranged as
$$2x + 30 = 2\sqrt{(x+39)(x+7)}$$
Dividing by 2:
$$x + 15 = \sqrt{(x+39)(x+7)}$$
Square again
$$x^2 + 30x + 225  = (x+39)(x + 7)$$
$$= x^2 + 46x + 273$$
So your equation is the same as
$$30x + 225 = 46x + 273$$
So $16x = -48$ and therefore $x = -3$.
When squaring equations always plug in your answer to make sure it's the actual solution. And in this case it is.
</s>