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

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Wed Feb 23, 2022 4:09 am

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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

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Source: Beat The GMAT — Problem Solving | Wed Feb 23, 2022 4:09 am

regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

Re: How many roots does the equation $\sqrt{x^2+1}+\sqrt{x^2+2}=2$ have?

by regor60 » Wed Feb 23, 2022 5:35 am

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

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