I'm having trouble with a problem that involves square roots, and I'm hoping that someone here can help me figure it out.
The problem states that for which real numbers $x$ does the equation $\sqrt{x^2-1}=\sqrt{x+1}\cdot\sqrt{x-1}$ hold true? The answer choices are:
(a) all real numbers
(b) all real numbers such that $|x|\geq1$
(c) all real numbers such that $x\geq1$
(d) none of the above
Here's what I've tried so far: I squared both sides of the equation to get $x^2-1=(x+1)(x-1)=x^2-1$. This means that the equation holds true for all real numbers $x$.
However, I'm not sure if my solution is correct. Can someone please confirm if my answer is correct or if I made a mistake somewhere? Any guidance or suggestions would be greatly appreciated.
Thank you in advance!
I can only assume that complex numbers are forbidden, and we treat the square root of a negative as “bad!”. The question is still interesting if $\Bbb C$ is involved though.
It seems to me the question-asker is really just trying to get you to think about when things are defined. Hint: you’ve shown that, so long as everything is defined, the equation should hold. So what conditions do we need on $x$ to ensure those square roots exist?