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If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y?

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Bunuel
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If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   01 May 2021, 07:45
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If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?


(A) \(−2\)

(B) \(−\sqrt{2}\)

(C) \(−1\)

(D) \(1\)

(E) \(\sqrt{2}\)
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B
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Bunuel
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Re: If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   09 May 2021, 02:00
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Bunuel wrote:
If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?


(A) \(−2\)

(B) \(−\sqrt{2}\)

(C) \(−1\)

(D) \(1\)

(E) \(\sqrt{2}\)


\(y(y-2x)=2- x^2\);

\(y^2-2xy=2-x^2\);

\(y^2-2xy+x^2=2\);

\((x-y)^2=2\).

Since x<y (x-y<0), then \(x-y=-\sqrt{2}\).

Answer: B.

Hope it's clear.
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Re: If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   01 May 2021, 07:54
Bunuel wrote:
If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?


(A) \(−2\)

(B) \(−\sqrt{2}\)

(C) \(−1\)

(D) \(1\)

(E) \(\sqrt{2}\)



Explanation: y(y-2x)=2 - x^2

y^2 - 2xy=2 - x^2

x^2 + y^2 -2xy=2

(x-y)^2 = 2,

x-y= -\sqrt{2} (given x<y ,thus x-y <0)

Hence B is the correct answer.
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Re: If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   01 May 2021, 07:54
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Bunuel wrote:
If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?


(A) \(−2\)

(B) \(−\sqrt{2}\)

(C) \(−1\)

(D) \(1\)

(E) \(\sqrt{2}\)



x<y or x-y<0, so our answer has to be negative. You can eliminate the positive values.


\(y(y − 2x) =2 − x^2\)

\(y^2− 2xy=2 − x^2\)

\(x^2-2xy+y^2=2\)

\((x-y)^2=2\)

\(x-y=\sqrt{2}=-\sqrt{2}\)

But x-y<0, so answer is B
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Re: If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   01 May 2021, 08:01
Bunuel wrote:
If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?


(A) \(−2\)

(B) \(−\sqrt{2}\)

(C) \(−1\)

(D) \(1\)

(E) \(\sqrt{2}\)


\(y(y − 2x) =2 − x^2\)
\(y^2 - 2xy = 2 - x^2\)
\(x^2 - y^2 -2xy = 2\)
\((x-y)^2 = 2\)
Taking square roots both sides
\(|x-y| = \sqrt{2}\)
\(x - y = -\sqrt{2}\) (As, y < x)
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Re: If x < y, and y(y − 2x) = 2 − x^2, what is the value of x − y? [#permalink] New post   01 May 2021, 08:14
Asked: If \(x < y\), and \(y(y − 2x) =2 − x^2\), what is the value of \(x − y\)?

\(y(y − 2x) =2 − x^2\)
\(y^2 - 2xy =2 − x^2\)
\(x^2 + y^2 - 2xy =2\)
\((x-y)^2 =2\)
\(x-y =-\sqrt{2}\); since x<y

IMO B
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Re: If x < y, and y(y 2x) = 2 x^2, what is the value of x y? [#permalink] New post   09 Jun 2023, 23:45
Why are we taking the answer as - sqrt 2. It could also be +ve. Please help me know the reason behind it.
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Re: If x < y, and y(y 2x) = 2 x^2, what is the value of x y? [#permalink] New post   09 Jun 2023, 23:47
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shikha_0203 wrote:
Why are we taking the answer as - sqrt 2. It could also be +ve. Please help me know the reason behind it.


This is explained. You just have to read the solutions carefully:

\((x-y)^2=2\).

Since x < y (x - y < 0), then \(x-y=-\sqrt{2}\).
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Re: If x < y, and y(y 2x) = 2 x^2, what is the value of x y? [#permalink]
09 Jun 2023, 23:47
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