<s>[INST] Solve $\sqrt{x+4}-\sqrt{x+1}=1$ for $x$ Can someone give me some hints on how to start solving $\sqrt{x+4}-\sqrt{x+1}=1$ for x?
Like I tried to factor it expand it, or even multiplying both sides by its conjugate but nothing comes up right.
 [/INST] Let $a = \sqrt{x+4}$ and $b=\sqrt{x+1}$. So that $a^2 = x + 4$ and $b^2 = x + 1$.
From the given equation,
$$\sqrt{x+4} - \sqrt{x+1}=1 \implies a - b = 1 \ \ \  \text{and} \ \ \ a^2-b^2=3.$$
So we have that,
$$a^2-b^2 =(a-b)(a+b)=1 \cdot(a+b)=3 \implies a+b=3.$$
We see that
$$(a+b)+(a-b) = 2a.$$
Also,
$$(a+b)+(a-b)=3+1=4.$$
Therefore,
$$2a=4 \implies a=2 \implies \sqrt{x+4}=2.$$
Solving for $x$,
$$x+4=4 \implies x=0.$$
</s>