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Pls help :How many roots does this equation have?

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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.
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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

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