<s>[INST] Simplest way to solve radical equation $\sqrt{3x+1}-\sqrt{x+4}=1$ I have the following equation:
$$\sqrt{3x+1}-\sqrt{x+4}=1$$
I can get the answer $x=5$ through tedious and long algebraic manipulation with quite a few extraneous solutions. It's not elegant. Is there a simple, straightforward way to solve this equation?
 [/INST] If we introduce the variables 
\begin{align}
y^2&=&3x+1\tag1\\
z^2&=&x+4\tag2\\
y-z&=&1\tag3\end{align}
we get an extraneous solution, but it is easy to dismiss and this makes the calculations a bit neater.  Subtract three times $(2)$ from $(1)$
\begin{align}
y^2-3z^2=-11\\\
\end{align}
using $(3)$
\begin{align}
(z+1)^2-3z^2+11=-2z^2+2z+12=0\\
\end{align}
$$\Leftrightarrow$$
\begin{align}
z^2-z-6=0\\
\end{align}
There are two solutions to this $z = -2$ and $z = 3$. We dismiss the first since $z$ represents a square root and we don't want this to be negative. The second leads to $x=5$, $y=4$ and $z=3$.
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