Question

If x, y are integral solutions of $2x^2 - 3xy - 2y^2 = 7$, then the value of |x + y| is

• A. 5
• B. 6
• C. 3
• D. 4

Answer 1

Answer: (D)

4

Answer 2

The given equation is $2x^2 - 3xy - 2y^2 = 7$. We need to find integral solutions $(x, y)$. The left side of the equation can be factored. We can treat it as a quadratic in $x$: $2x^2 - (3y)x - 2y^2$. We look for factors of the form $(ax+by)(cx+dy)$. The product of the $x^2$ coefficients must be 2, so $ac=2$. Possible pairs for $(a, c)$ are $(1, 2)$ or $(2, 1)$ or $(-1, -2)$ or $(-2, -1)$. The product of the $y^2$ coefficients must be -2, so $bd=-2$. Possible pairs for $(b, d)$ are $(1, -2), (-1, 2), (2, -1), (-2, 1)$. The coefficient of $xy$ is $ad+bc = -3$.

Let's try the form $(x+by)(2x+dy)$. $ad+2b = -3$. If $(b, d) = (1, -2)$, then $1(-2) + 2(1) = -2 + 2 = 0 \ne -3$. If $(b, d) = (-1, 2)$, then $1(2) + 2(-1) = 2 - 2 = 0 \ne -3$. If $(b, d) = (2, -1)$, then $1(-1) + 2(2) = -1 + 4 = 3 \ne -3$. If $(b, d) = (-2, 1)$, then $1(1) + 2(-2) = 1 - 4 = -3$. This works. So the factors are $(x - 2y)(2x + y)$.

Let's verify the factorization: $(x - 2y)(2x + y) = x(2x+y) - 2y(2x+y) = 2x^2 + xy - 4xy - 2y^2 = 2x^2 - 3xy - 2y^2$. The factorization is correct. So the equation is $(x - 2y)(2x + y) = 7$. Since $x$ and $y$ are integers, the factors $(x - 2y)$ and $(2x + y)$ must be integer factors of 7. The integer factors of 7 are $1, -1, 7, -7$. We have four possible cases for the values of the factors:

Case 1: $x - 2y = 1$ and $2x + y = 7$. Multiply the second equation by 2: $4x + 2y = 14$. Add the first equation to this: $(x - 2y) + (4x + 2y) = 1 + 14 \implies 5x = 15 \implies x = 3$. Substitute $x = 3$ into $2x + y = 7$: $2(3) + y = 7 \implies 6 + y = 7 \implies y = 1$. So $(x, y) = (3, 1)$ is an integral solution. For this solution, $|x + y| = |3 + 1| = |4| = 4$.

Case 2: $x - 2y = 7$ and $2x + y = 1$. Multiply the second equation by 2: $4x + 2y = 2$. Add the first equation to this: $(x - 2y) + (4x + 2y) = 7 + 2 \implies 5x = 9 \implies x = 9/5$. This is not an integer, so there is no integral solution in this case.

Case 3: $x - 2y = -1$ and $2x + y = -7$. Multiply the second equation by 2: $4x + 2y = -14$. Add the first equation to this: $(x - 2y) + (4x + 2y) = -1 + (-14) \implies 5x = -15 \implies x = -3$. Substitute $x = -3$ into $2x + y = -7$: $2(-3) + y = -7 \implies -6 + y = -7 \implies y = -1$. So $(x, y) = (-3, -1)$ is an integral solution. For this solution, $|x + y| = |-3 + (-1)| = |-4| = 4$.

Case 4: $x - 2y = -7$ and $2x + y = -1$. Multiply the second equation by 2: $4x + 2y = -2$. Add the first equation to this: $(x - 2y) + (4x + 2y) = -7 + (-2) \implies 5x = -9 \implies x = -9/5$. This is not an integer, so there is no integral solution in this case.

The integral solutions are $(3, 1)$ and $(-3, -1)$. For both solutions, the value of $|x + y|$ is 4.