If x ¤ y = (x + y)^2 - (x - y)^2. Then √5 ¤ √5 =

A. 0
B. 5
C. 10
D. 15
E. 20
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Hi
if x ¤ y = (x + y)^2 - (x - y)^2 then √5 ¤ √5 = (√5+√5)^2 - (√5-√5)^2
let solve the equation:
\(√5 ¤ √5 = (√5+√5)^2 - (√5-√5)^2\)
\(=>√5 ¤ √5 = (√5+√5)^2\)
\(=>√5 ¤ √5 = (√5)^2 + 2 * √5 * √5 +(√5)^2\)
\(=>√5 ¤ √5 = 5 + 2 * 5 + 5\)
\(=>√5 ¤ √5 = 5+10+5\)
\(=>√5 ¤ √5 = 10+10\)
\(=>√5 ¤ √5 = 20\)

=> answer is E (20)

This is a great questions that typifies what I love about most GMAT math questions: they can almost always be solved in more than 1 way.

One option is to plug the values into the "recipe" and get...
√5 ¤ √5 = (√5 + √5)² - (√5 - √5)²
= (2√5)² - (0)²
= (2√5)(2√5)
= 4√25
= 20
= D

Another option is to simplify the recipe before plugging in the values
x ¤ y = (x + y)² - (x - y)²
= (x² + 2xy + y²) - (x² - 2xy + y²)
= 4xy
So, x ¤ y = 4xy
NOW plug in the values to get....
√5 ¤ √5 = 4(√5)(√5)
= 4√25
= 20
= D

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\(If x ¤ y = (x + y)^2 - (x - y)^2. Then √5 ¤ √5 =\)

Another way to simplify is to realize that this is a difference of squares pattern. Really, it's a difference of two terms pattern if I can call it that.

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

So, you can spell it out but if you realize, the first terms are positive when factoring, and the second set of terms has just one negative. You'll really end up with \((2*\sqrt{5})\) \((2*\sqrt{5})\)