a=x+y and b=x-y. If a^2=b^2, what is the value of y?

Hi Mo2men,

There are a couple of issues here:

First, √9 has just ONE solution: +3

The equation X^2 = 9 has TWO solutions: +3 and -3.... but we have to work with what we were given and we were given RADICALS, so there is no negative option.

Second, knowing that the radical has just one solution, X = 0 is NOT an option in Fact 2, since we need the result of the calculation to be > 0. By extension, Y MUST = 0.

GMAT assassins aren't born, they're made,
Rich

Hello Bunuel

But square root of 4 is +/-2 correct? And -2 < 0
Can u please explain why u said square root of integer cannot b less than 0?

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Hi All,

This DS question is rooted in Classic Quadratics and Radical rules. There's a heavy "logic" component to this prompt, and you'll have to do a certain amount of math to get to the solution...

We're given a number of facts to work with:
1) A = X+Y
2) B = X-Y
3) A^2 = B^2

We're asked for the value of Y.

Before dealing with the two Facts, I'm going to take a moment to simplify the information in the prompt.

A^2 = (X+Y)^2 = X^2 + 2XY + Y^2
B^2 = (X-Y)^2 = X^2 - 2XY + Y^2

A^2 = B^2.....so......

X^2 + 2XY + Y^2 = X^2 - 2XY + Y^2

If we cancel out the terms, we're left with...
2XY = -2XY

There are only 3 ways for this equation to exist....
X = 0 OR Y = 0 OR BOTH = 0

This is an important restriction that will impact the answer to this question.

Fact 1: √X + √Y > 0

Since radicals are either POSITIVE or ZERO, either X or Y could = 0 (but not both).
IF....X=1, Y=0 the answer to the question is 0
IF....X=0, Y=1 the answer to the question is 1
Fact 1 is INSUFFICIENT

Fact 2: √X − √Y > 0

Here, the DIFFERENCE is > 0, so the only way for that difference to be POSITIVE, given ALL of the restrictions we've already discussed, is for the X to be POSITIVE and the Y to be 0. Here, the answer to the question is ALWAYS 0.
Fact 2 is SUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Hi Rich,
Nice explanation and powerful 'TESt It' methodology. I reached the same result: X = 0 OR Y = 0 OR BOTH = 0 but I was afraid that I'm mistaken because of zeros in the result so I applied 'Testing it' in the equation but it got so complicated with many variables and situations.


However, I want to ask about Fact 2. If I choose X=0 & Y= 9 for easy roots example. So The result will be the following:
√X − √Y > 0 then Fact 2: √0 − √9 > 0------> .... √9 will be either 3 or -3. If it is 3, then 0-3>0 so not satisfying the condition but if Y=-3, then 0- (-3)= 3>0 so it is OK with condition. Therefore, Y will always any positive number. So Fact 2 should be insufficient as there many values for Y.

What is wrong in the above?

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

\(a^2 = b^2\)
\(⇔ (x+y)^2 = (x-y)^2\)
\(⇔ x^2 + 2xy + y^2 = x^2 - 2xy + y^2\)
\(⇔ 2xy = -2xy\)
\(⇔ 4xy = 0\)
\(⇔ x = 0\) or \(y = 0\)

Condition 1)
\(x = 1, y = 0\) or \(x = 0, y = 1\)
Since we don't have a unique solution, condition 1) is not sufficient.

Condition 2)
\(\sqrt{x} - \sqrt{y} > 0\)
\(⇔ \sqrt{x} > \sqrt{y}\)
\(⇔ x > y ≥ 0\)
With the original condition, we have \(y = 0\).
Condition 2) is sufficient.

Therefore, B is the answer.

here also as per maths rules, we can't have power to zero. Means 0^1, 0^0.5, 0^3...etc are not possible. And hence, we can't say y=0 from (B). So, correct answer should be (E)

That's not true.

Zero to any positive power equals zero. For example, 0^3 = 0*0*0 = 0.

Zero to any negative power is undefined. For example, 0^(-5) = 1/0^5 = 1/0, which is undefined because we cannot divided by zero.

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