<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] HINT:
As $(x+4)-(x+1)=3 \ \ \ \ \ $
$\implies (\sqrt{x+4}-\sqrt{x+1})(\sqrt{x+4}+\sqrt{x+1})=3$
$$\text{As }\sqrt{x+4}-\sqrt{x+1}=1\ \ \ \ \ (1)$$
$$\implies  \sqrt{x+4}+\sqrt{x+1}=3\ \ \ \ \ (2)$$
Add/subtract  $(1)$ and $(2),$ then square
Generalization :
$$\text{As }(ax+b)-(ax+c)=b-c$$
$$\text{If }\sqrt{ax+b}-\sqrt{ax+c}=d \ \ \ \ \ (1) $$
$$\text{As } (ax+b)-(ax+c)=(\sqrt{ax+b}-\sqrt{ax+c})(\sqrt{ax+b}+\sqrt{ax+c})$$
$$\implies \sqrt{ax+b}+\sqrt{ax+c}=\frac{b-c}d\ \ \ \ \ (2)$$
Add/subtract $(1)$ and $(2),$ then square
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