I am sooo lost on this question. I solved it the same as TheSiuation. I am do not understand your method below. Although I am sure it is pbvious to everyone but please clarify the rule, therum or postulate that you followed to solve the equation. Thanks in advacne.

TheSituation wrote:

I got A but solved thru a different (less clever) method. I think I may have answered correct in spite of myself though, can someone tell me if my solution is mathmatically sound or if I arrived at the correct solution by luck.

Begin by squaring both sides

(x^2+1) + (x^2 +2) = 4

drop brackets and solve

2x^2=1

x^2=1/2

x = sq root of 1/2

therefore no real roots, therefore A

feedback?

This would be the longer way, plus you'll need to square twice not once, as you made a mistake while squaring first time.

\((a+b)^2=a^2+2ab+b^2\), so when you square both sides you'll get: \(x^2+1+2\sqrt{(x^2+1)(x^2 +2)}+x^2+2= 4\) --> \(2\sqrt{(x^2+1)(x^2 +2)}=1-2x^2\). At this point you should square again --> \(4x^4+12x^2+8=1-4x^2+4x^4\) --> \(16x^2=-7\), no real \(x\) satisfies this equation.

There is one more problem with your solution:

You've got (though incorrectly): \(x^2=\frac{1}{2}\) and then you concluded that this equation has no real roots, which is not right. This quadratic equation has TWO real roots: \(x=\frac{1}{\sqrt{2}}\) and \(x=-{\frac{1}{\sqrt{2}}}\). Real roots doesn't mean that the roots must be integers, real roots means that roots must not be complex numbers, which I think we shouldn't even mention as GMAT deals ONLY with real numbers. For example \(x^2=-1\) has no real roots and for GMAT it means that this equation

has no roots, no need to consider complex roots and imaginary numbers.

Hope it's clear.