Kaplan DS Question

[willshu]

If x + y^2 = ( x + y^2 ) ^2 , what is the value of y ?

(1) x = y^2

(2) xy^2 = 0

My question concerns the second part of the explanation. The explanation indicates that because xy^2 = 0, either x or y must be equal to zero, which makes sense. However, the explanation simply assumes x is zero and not y. How can you do that? Wouldn't you get a different answer if you substitute y = 0 instead of x = 0?

Thanks


[Show/hide explanation]

The question gives you the equation x + y2 = ( x + y2 ) 2 and asks you for the value of y . We could try to rearrange this equation; however, since it is relatively complicated, let's look at the statements to see if it might be a little faster to work with the equation of the question stem and another equation.

Let's look at statement (1), x = y2 . Let's substitute y2 for x into the equation x + y2 = ( x + y2 ) 2 of the question stem. Then y2 + y2 + ( y2 + y2 ) 2 . Solve this for the value or values of y . Then 2 y2 + (2 y2 ) 2 , 2 y2 + 2 2 ( y 2 × 2 ), and 2 y2 = 4 y4 . Next, 4 y4 - 2 y2 = 0. Now factor out a 2 y2 from the left side of this equation. Then 2 y2 (2 y2 - 1) = 0. When the product of a group of numbers is 0, at least one of the numbers must be 0. So either y2 = 0 or 2 y2 - 1 = 0. If y2 = 0, then y = 0. If 2 y2 - 1 = 0, then 2 y2 = 1, y2 = , and y = - or y = - .

So from statement (1) we know that y must be - , 0, or . Since we can't determine a single value for y , statement (1) is insufficient. Eliminate (A) and (D).

Now let's look at statement (2), xy2 = 0. We've already mentioned that when the product of a group of numbers is 0, at least one of the numbers must be 0. So either x = 0 or y = 0 Suppose that x = 0. Substitute 0 for x into the equation x + y2 = ( x + y2 ) 2 . Then 0 + y2 = (0 + y2 ) 2 , y2 = ( y2 ) 2 , y2 = y 2 × 2 , and y2 = y4 . Next, y4 - y2 = 0, y2 ( y2 - 1) = 0, and y2 ( y + 1)( y - 1) = 0. If y2 = 0, then y = 0. If y + 1 = 0, then y = - 1. If y - 1 = 0, then y = 1. So if x = 0 , then y must be - 1, 0, or 1. Since we cannot determine a single value for y , statement (2) is insufficient.

Now take the statements together. From statement (1) we know that y must be - , 0, or . From statement (2) we know that y must be - 1, 0, or 1. The only value of y that is possible with each of the two statements is 0. So y = 0. The two statements taken together are sufficient and (C) is correct.

[Talkativetree]

If x + y^2 = ( x + y^2 ) ^2 , what is the value of y ?

(1) x = y^2

(2) xy^2 = 0

Solution first, then I'll look at your question.

(1) x = y^2
x + y^2 = ( x + y^2 ) ^2
y^2 + y^2 = ( y^2 + y^2 ) ^2
2y^2 = (2y^2)^2
y x (square root of 2) = 2y^2
(square root of 2)/2 = y

HOWEVER, at the start, you SHOULD notice that you have a constant x a variable = another constant x the same variable. another possible answer is y=0. So (1) is insufficient.

(2) xy^2 = 0
This says that either x or y = 0. insufficient on it's own.

at this point, answer is either E or C.

When you combine (1) and (2), you should see that out of the two possible answers for y from (1), (square root of 2)/2 or 0, there is only one that also satisfies (2).

both x and y are 0 I believe.

If you are having problems with that, try and calculate out x=y^2, with the different values of y, and then plug those into (2).

Okay, about your question about the explanation, the guide doesn't actually say that y couldn't equal 0. The guide is only doing enough work to find out whether or not it is sufficient, and by starting with the trying x=0 first (because you are trying to find the value for y, so you shouldn't start off by assuming it's 0), the explanation sees that y could be any value, thus it's insufficient. The thing is, y COULD be O, but it could also be 1, 5, -232, or 4.

did I clarify it well enough?

[brick2009]

A.) X = y^2

if you substitute this in the main eqn.
after simplification you end up with:

2.y^2 = 4.y^4
=> y^2 = 1/2
=> Insufficient as you get two values of Y (a-d- OUT)

B.) X.y^2 = 0

you know why B wont help.

NOW:

Taking A & B Together:

take the main equation and expand it and simplify it:

x+y^2 = x^2+2.x.Y^2 + y^4

x.y^2 = 0, and x = y^2

So that reduces to : 1 = Y^2 = > y = +/- 1


So i am getting : E as the answer.

What is OA??? (did i miss something)

[Talkativetree]

if you look at the end of the explanation, the answer is C.

what you missed is that basically x(y^2)=0, which means either x or y MUST equal 0.

When adding in the equation x=y^2 into the mix, this means that, given one of the two variables must = 0, the only solution is that both equal 0.

[KapTeacherEli]

The step that everyone has missed to far is analyzing the question stem. Since we have a complex term (x+y^2) equal to it's own square, we know that one of two things is true: x+y^2 = 1 or 0, the only two numbers whose values are constant when squared.

1) x = y^2
Remember our deduction about the value of x + y^2. That, combined with this statement, means either x and y^2 both equal 0, or x and y^2 both equal 1/2. y is either 0 or (root 2) / 2; insufficient.

(2) xy^2 = 0 If we assume x = 0, we have two values for y that satisfy the equation in the stem--y could be 0 or 1. Insufficient.

Combined, however, there is only one value that satisfies both: y = 0. Answer is C.

And, to answer your question, assuming y = 0 doesn't work because we can't assume what the question was asking us! Remember, we need to know IF y is 0, so assuming it is would be counterproductive!