If x + y = 18 and xy = 72 for real numbers x and y, determine x^2 + y^2 using the standard sum-of-squares identity.

Introduction / Context:Instead of solving a quadratic to find x and y individually, the identity linking the square of the sum to the sum of squares makes it trivial to compute x^2 + y^2 directly from x + y and xy. This is a common, time-saving technique in aptitude tests.

Given Data / Assumptions:

• x + y = 18
• xy = 72
• Find x^2 + y^2.

Concept / Approach:Use (x + y)^2 = x^2 + y^2 + 2xy. Rearranging gives x^2 + y^2 = (x + y)^2 − 2xy. Plug in the known values and compute.

Step-by-Step Solution:Compute (x + y)^2 = 18^2 = 324.Apply identity: x^2 + y^2 = 324 − 2*72 = 324 − 144 = 180.

Verification / Alternative check:Optionally, solve the quadratic t^2 − 18t + 72 = 0 to get t = 12 and 6; then x^2 + y^2 = 12^2 + 6^2 = 144 + 36 = 180, verifying the identity method.

Why Other Options Are Wrong:

• 120, 90, 144: These come from misapplying the identity or errors with the factor 2 in 2xy.
• “Cannot be determined”: Incorrect because the identity gives a unique value without needing the individual numbers.

Common Pitfalls:Mixing up (x + y)^2 with x^2 + y^2; forgetting to multiply xy by 2.

Final Answer:180