Question

\(x\) and \(y\) are non-zero integers. Compare the following: \[ \text{Quantity A: } (x - y)^2 \quad \quad \text{Quantity B: } (x + y)^2 \]

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The answer cannot be determined from the information given.

Hint:

When comparing squared terms, expand and subtract them to see the dependence on the cross term. Here, the result depended on the sign of \(xy\).

Answer 1

The Correct Option is D

Step 1: Expand both quantities.
\[ (x-y)^2 = x^2 - 2xy + y^2, \quad (x+y)^2 = x^2 + 2xy + y^2 \]
Step 2: Compare.
The only difference is the middle term: \[ (x+y)^2 - (x-y)^2 = 4xy \]
Step 3: Analyze cases.
- If \(xy>0\) (both \(x\) and \(y\) same sign), then \((x+y)^2>(x-y)^2\).
- If \(xy<0\) (opposite signs), then \((x-y)^2>(x+y)^2\).
- If \(xy = 0\), it contradicts the condition that \(x,y \neq 0\). So the relationship cannot be determined without knowing the signs of \(x\) and \(y\).
Final Answer: \[ \boxed{\text{The answer cannot be determined from the information given.}} \]