<s>[INST] In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions? In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?
I tried shifting the second term to the rhs and squaring.Even after that i'm left with square roots.No idea how to proceed.Help!
 [/INST] Hint:
Notice that $$x+3-4\sqrt{x-1}=x-1-4\sqrt{x-1}+4=(\sqrt{x-1}-2)^2$$
and
$$x+8-6\sqrt{x-1}=x-1-6\sqrt{x-1}+9=(\sqrt{x-1}-3)^2=(3-\sqrt{x-1})^2$$
After, you can try by cases.
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