<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] $$
\sqrt{x + 39} - \sqrt{x + 7} = 4 \quad \Rightarrow \quad (\sqrt{x + 39})^2 = (\sqrt{x + 7} + 4)^2 \quad \Rightarrow 
$$
$$
x + 39 = x + 7 + 8\sqrt{x + 7} + 16 \quad \Rightarrow \quad 8\sqrt{x + 7} = 16 \quad \Rightarrow \quad x = -3
$$
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