Evaluate the symmetric expression for nearby integers: If x = 997, y = 998, and z = 999, compute the value of x^2 + y^2 + z^2 − xy − yz − zx.
Aptitude
Elementary Algebra
Difficulty: Easy
Choose an option
-
A3
-
B9
-
C16
-
D4
Answer
Correct Answer: 3
Explanation
Introduction / Context:This problem tests recognition of a standard symmetric identity that simplifies expressions of the form x^2 + y^2 + z^2 − xy − yz − zx. Using the identity avoids large-number squaring and reduces the task to simple differences.
Given Data / Assumptions:
- x = 997
- y = 998
- z = 999
- Target expression: x^2 + y^2 + z^2 − xy − yz − zx
Concept / Approach:Use the identity: x^2 + y^2 + z^2 − xy − yz − zx = 1/2 * [ (x − y)^2 + (y − z)^2 + (z − x)^2 ]. This identity is derived by expanding the squares and collecting terms, and it dramatically simplifies computation when the numbers are close together.
Step-by-Step Solution:
Compute pairwise differences: x − y = 997 − 998 = −1.y − z = 998 − 999 = −1.z − x = 999 − 997 = 2.Squares: (−1)^2 + (−1)^2 + (2)^2 = 1 + 1 + 4 = 6.Apply the identity: 1/2 * 6 = 3.Verification / Alternative check:
Direct computation with large squares would be tedious; the identity provides a clean shortcut and the result is a small integer, which is consistent with the tiny gaps between x, y, and z.Why Other Options Are Wrong:
- 9, 16, 4: These do not match the identity-based calculation using true differences −1, −1, and 2.
Common Pitfalls:
- Forgetting the 1/2 factor in the identity.
- Sign mistakes in differences, though squares remove signs; the only risk is miscomputing 2^2.
Final Answer:
3