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x and y are integers such that 4x^2 – y^2 +4x + 4y – 3 =

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x and y are integers such that 4x^2 – y^2 +4x + 4y – 3 =

by Max@Math Revolution » Thu Jun 14, 2018 1:36 am

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[GMAT math practice question]

x and y are integers such that 4x^2 - y^2 +4x + 4y - 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number
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by GMATGuruNY » Thu Jun 14, 2018 3:03 am
Max@Math Revolution wrote:[GMAT math practice question]

x and y are integers such that 4x^2 - y^2 +4x + 4y - 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number
Let x=1.
Substituting x=1 into 4x² - y² + 4x + 4y - 3 = 0, we get:
4(1²) - y² + (4*1) + 4y - 3 = 0
4 - y² + 4 + 4y - 3 = 0
- y² + 4y + 5 = 0
y² - 4y - 5 = 0
(y-5)(y+1) = 0
y=5 or y=-1.

If y=-1, then A, C and E are not true.
If y=5, then D is not true.

The correct answer is B.
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by Brent@GMATPrepNow » Thu Jun 14, 2018 5:13 am
Max@Math Revolution wrote:[GMAT math practice question]

x and y are integers such that 4x² - y² + 4x + 4y - 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number
Let's examine the expression in parts
4x² is EVEN for all integer values of x
y² not sure
4x is EVEN for all integer values of x
4y is EVEN for all integer values of y
3 is ODD
0 is EVEN

So, we get: EVEN - y² + EVEN + EVEN - ODD = EVEN
Simplify left side: ODD - y² = EVEN
This mean y² is ODD
If y² is odd, then y MUST BE ODD

Answer: B

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Brent
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by Max@Math Revolution » Sun Jun 17, 2018 5:15 pm
=>

4x^2 - y^2 +4x + 4y - 3 = 0
=> 4x^2 + 4x + 1 - y^2 + 4y - 4 = 0
=> 4x^2 + 4x + 1 = y^2 - 4y + 4
=> (2x+1)^2 = (y-2)^2
Since 2x + 1 is an odd integer, (y-2)^2 is an odd integer and y-2 is an odd integer.
Thus, y is also an odd integer.

Therefore, the answer is B.
Answer : B
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by Jeff@TargetTestPrep » Sun Jun 17, 2018 7:06 pm
Max@Math Revolution wrote:[GMAT math practice question]

x and y are integers such that 4x^2 - y^2 +4x + 4y - 3 = 0. Which of following is true?

A. y is an even number.
B. y is an odd number.
C. y is positive.
D. y is negative.
E. y is a prime number
Simplifying the equation, we have:

4x^2 + 4x - y^2 + 4y = 3

Even + Even - y^2 + Even = 3

Since Even + Even + Even = Even we have:

Even - y^2 = 3

Thus, y^2, and also y, must be odd.

Answer: B

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