<s>[INST] For $x$ and $k$ real numbers, for what values of $k$ will the graphs of $f(x)=-2\sqrt{x+1}$ and $g(x)=\sqrt{x-2}+k$ intersect? 
For $x$ and $k$ real numbers, for what values of $k$ will the graphs of $f(x)=-2\sqrt{x+1}$ and $g(x)=\sqrt{x-2}+k$ intersect?

I tried to make an equation of them, but I’m stuck with the two variables and I couldn’t solve it. Much appreciation.
We didn’t do calculus yet..
 [/INST] Hint: You must solve the equation
$$\sqrt{x-2}+k=-2\sqrt{x+1}$$ for $$x\geq 2$$ Writing this equation in the form
$$-k-\sqrt{x-2}=2\sqrt{x+1}$$ then it must be
$$k^2+2\geq x$$ and you can square it.
After squaring one times we get
$$2k\sqrt{x-2}=3x+6-k^2$$
Squaring again we get
$$-k^4+10 k^2 x+4 k^2-9 x^2-36 x-36=0$$
Solving this we get
$$k\leq -2 \sqrt{3}\land x=\frac{1}{9} \left(5
   k^2-18\right)-\frac{4}{9} \sqrt{k^4-9 k^2}$$
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