Pls help :How many roots does this equation have?

This topic has expert replies
User avatar
Master | Next Rank: 500 Posts
Posts: 345
Joined: Sun Dec 21, 2008 10:57 pm
Thanked: 6 times
Followed by:3 members
How many roots does this equation have?

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

0
1
2
3
4

User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Mon Sep 08, 2014 2:40 am
himu wrote:How many roots does this equation have?

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

0
1
2
3
4
Every test-taker should know that √2≈1.4.

Since the square of a value cannot be negative, the least possible value of x² is 0.
Substituting x²=0 into √(x²+1) + √(x²+2) = 2, we get:
√(0²+1) + √(0²+2) = 2
√1 + √2 = 2.
2.4 = 2.

Doesn't work:
The lefthand side is greater than the righthand side.
Since there is no way to decrease the value of the lefthand side, the equation has no solutions.

The correct answer is A.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Mon Sep 08, 2014 11:13 am
Hi himu,

Sometimes a weird "looking" calculation isn't really about doing a calculation, it's about estimating a value that is smaller or bigger.

We know that X^2 is greater than or equal to 0.

We also know that √1 = 1

So when we add up the two radicals...

The first radical has to be AT LEAST 1 (since we COULD have 0+1 or something bigger).
The second radical is GREATER THAN 1 (since we COULD have 0+2 or something bigger).

These two values, when summed, will be GREATER THAN 2, but the equal states that the sum has to EQUAL 2, so we know that there is no solution to the given equation.

Final Answer: A

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image