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.
(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.
