<s>[INST] Show that the equation $\sqrt{x+5}+\sqrt{x}=2$ has no real roots 
Show that the equation $$\sqrt{x+5}+\sqrt{x}=2$$ has no real roots.

What is the fastest approach to solve the problem? We have $$\sqrt{x+5}+\sqrt{x}=2 \iff \sqrt{x+5}=2-\sqrt{x}$$$$D_x:\begin{cases}x+5\ge0\\x\ge0\\2-\sqrt{x}\ge0\end{cases}\iff x\in[0;4].$$$$x+5=4-4\sqrt{x}+x \text{ ...}$$ We will get that the equation has no roots.
Can I ask you what steps should I follow when I am solving a radical equation with 2 square roots?
 [/INST] $$\sqrt{x+5}+\sqrt{x}=2\implies$$
$$\sqrt{x+5}=2-\sqrt{x}\implies$$
$$(\sqrt{x+5})^2=(2-\sqrt{x})^2\implies$$
$$x+5=4+x-4\sqrt{x}\implies$$
$$1=-4\sqrt{x}$$
which is impossible since $1>0$ and $-4\sqrt{x}\le0$.
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