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

Question

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

quantumliner · Accepted Answer

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

niks18 · Answer

The expression can be simplified as  x Ф y = (\frac{x}{2}-y)^2 
therefore  = (\frac{186}{2}-91)^2 = 4 
Option  B

sashiim20 · Answer

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)... _________________ Please Press "+1 Kudos" to appreciate.

akshata19 · Answer

Good question to be done mentally

xФy=(x/2−y)^2=(93−91)^2=4

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BrentGMATPrepNow · Answer

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) ²
= (1/4) ²
= (1/4) ²
= (1/4) 
= 4

Answer: B

napolean92728 · Answer

Simplify the expression: 
Recognize that x2/4 - xy + y2 can be rewritten as (x/2 - y)2.
Substitute the values:

x = 186, y = 91
186 Ф 91 = (186/2 - 91)2

Calculate: 
186/2 = 93
93 - 91 = 2
22 = 4

Answer: 
186 Ф 91 = 4 ✅

ravi1522 · Answer

Hello,
x Ф y is x^2/4-x*y+y^2 now this can be written as (x/2-y)^2
so 186 Ф 91 (186/2 -91)
(93-91)^2=4

Hence answer is option B

Thanks

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

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