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An operation Ф is defined by the equation x Ф y = x²/4

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An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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An operation Ф is defined by the equation \(x Ф y = \frac{x^2}{4} - xy + y^2\) for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

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Re: An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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New post 02 Aug 2017, 11:32
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xФy=\(\frac{x^2}{4}−xy+y^2\) , can be simplified to \((\frac{x}{2} - y)^ 2\)

So, 186 Ф 91 = \((\frac{186}{2} - 91)^ 2\) = \((93 - 91)^2\) = \(2^2\) = 4

Answer is B
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Re: An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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New post 02 Aug 2017, 11:32
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GMATPrepNow wrote:
An operation Ф is defined by the equation \(x Ф y = \frac{x^2}{4} - xy + y^2\) for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

*kudos for all correct solutions


The expression can be simplified as \(x Ф y = (\frac{x}{2}-y)^2\)
therefore \(= (\frac{186}{2}-91)^2 = 4\)
Option \(B\)
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An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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New post 02 Aug 2017, 11:34
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GMATPrepNow wrote:
An operation Ф is defined by the equation \(x Ф y = \frac{x^2}{4} - xy + y^2\) for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

*kudos for all correct solutions


\(x\) \(Ф\) \(y = \frac{x^2}{4} - xy + y^2\) \(=> \frac{(x^2 - 4xy + 4y^2)}{4}\)

\(x\) \(Ф\) \(y = \frac{(x - 2y)^2}{4}\)

\(186\) \(Ф\) \(91 = \frac{(186 - 2*91)^2}{4}\)

\(186\) \(Ф\) \(91 = \frac{(186 - 182)^2}{4}\)

\(186\) \(Ф\) \(91 = \frac{4^2}{4}\)

\(186\) \(Ф\) \(91 = 4\)

Answer (B)...

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An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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New post 02 Aug 2017, 19:31
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Good question to be done mentally


xФy=(x/2−y)^2=(93−91)^2=4

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Re: An operation Ф is defined by the equation x Ф y = x²/4 [#permalink]

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New post 03 Aug 2017, 07:40
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GMATPrepNow wrote:
An operation Ф is defined by the equation \(x Ф y = \frac{x^2}{4} - xy + y^2\) for all numbers x and y. What is the value of 186 Ф 91?

A) 1
B) 4
C) 8
D) 16
E) 24

*kudos for all correct solutions


When we see that the expression x²/4 - xy + y² has two squares (x²/4 = (x/2)² and y² is obviously a square), we should start wondering whether the expression can be factored into the form (something)²

To make this factorization easier to see, let's factor out 1/4 first.
We get: x²/4 - xy + y² = (1/4)(x² - 4xy + 4y²)
= (1/4)(x - 2y)²

Perfect, we can now say that x Ф y = (1/4)(x - 2y)²
So, 186 Ф 91 = (1/4)[186 - 2(91)]²
= (1/4)[186 - 182]²
= (1/4)[4]²
= (1/4)[16]
= 4

Answer:

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Re: An operation Ф is defined by the equation x Ф y = x²/4   [#permalink] 03 Aug 2017, 07:40
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