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I don't think we need to do any algebra here: \(x^2 \geq 0\), so \(\sqrt{x^2+1} + \sqrt{x^2+2} \geq 1 + \sqrt{2}\), which is larger than 2. Hence no (real) solutions for x.

Please explain that how we have assumed \(x^2 \geq 0\) at the start of the problem and what is the theory behind this assumption. I have also solved with the long method as mentioned by Bunnel.

it doesn't have any solution beacuse left hand side is always greater than 2. the min value of left hand side is 2.414. which is greater than 2.
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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.

this is the kind of questions i had got in real GMAT and i was stumped...took 5 mins, yet no correct answer.. .now i am slowly understanding how to answer these kinds
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Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Tricky question. After 2 minutes I had to pick answer randomly and got this question wrong... I saw because x^2 >= 0, so the left side is always > 2. Very nice question.
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I don't think we need to do any algebra here: \(x^2 \geq 0\), so \(\sqrt{x^2+1} + \sqrt{x^2+2} \geq 1 + \sqrt{2}\), which is larger than 2. Hence no (real) solutions for x.

The solution explained by you is just awesome. Though I solved the problem but I use lot of algebra and took much time to solve the problem. Nice explanation
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