<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] What do you consider long and tedious.
I usually isolate one radical to one side by itself and square, repeat, keep track of signage and it usually works well
$\sqrt{3x+1}-\sqrt{x+4}=1$
$\sqrt{3x + 1} = 1 + \sqrt{x+4}$ [make note $3x+1 \ge 1; x+4 \ge 0$]
$3x + 1 = (1 + \sqrt{x+4})^2 = 1 + 2\sqrt{x+4} + x+4$
$2x -4 = 2\sqrt{x+4}$
$x -2 = \sqrt{x+4}$ [Make note $x-2 \ge 0$
$(x-2)^2 = x + 4$
$x^2 - 4x +4 = x + 4$
$x^2 -5x = 0$.
Quadratic equation or in this case simple factoring yields:
$x(x-5)= 0$
So $x = 0$ or $x = 5$.  As $0 - 2 < 0$ so that's out.  So $x = 5$.
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