How many roots does the equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) have?

[VJesus12]

How many roots does the equation \(\sqrt{x^2+1}+\sqrt{x^2+2}=2\) have?

A. 0
B. 1
C. 2
D. 3
E. 4

Answer: A

Source: GMAT Club Tests

[regor60]

The X^2 inside the radicals means the left side is always increasing with increasing X and symmetrical around the Y axis, which also means that the left side -2 is always increasing.

If the Y intercept is 0 or greater, then the curve doesn't intercept the X axis since the curve would be entirely above Y=0.

The Y intercept is found by setting X=0:

1+2^(1/2)-2 is about 1+1.4-2, which means a positive Y intercept so the answer is 0,A