If x + y = 12 and the product xy is equal to 32, what is the value of x^2 + y^2 expressed in simplest form using algebraic identities?

Introduction / Context:
This question checks knowledge of basic algebraic identities and how to use them to find expressions like x^2 + y^2 when you know x + y and xy. These types of problems are standard in aptitude tests and help build fluency with algebraic manipulation without solving for x and y individually.

Given Data / Assumptions:

• x + y = 12.
• xy = 32.
• We are required to compute x^2 + y^2.
• No further restrictions are placed on x and y beyond being real numbers.

Concept / Approach:
We use the identity (x + y)^2 = x^2 + 2xy + y^2. By rearranging this identity, we can express x^2 + y^2 in terms of (x + y)^2 and xy. This avoids solving for x and y directly. Specifically, x^2 + y^2 = (x + y)^2 - 2xy. Once we substitute the given values of x + y and xy, the computation becomes straightforward.

Step-by-Step Solution:
Start with the identity: (x + y)^2 = x^2 + 2xy + y^2.Rearrange to isolate x^2 + y^2: x^2 + y^2 = (x + y)^2 - 2xy.Substitute x + y = 12 into the formula: (x + y)^2 = 12^2 = 144.Substitute xy = 32: 2xy = 2 * 32 = 64.Therefore, x^2 + y^2 = 144 - 64 = 80.

Verification / Alternative check:
We can verify by finding specific values of x and y that satisfy x + y = 12 and xy = 32. These are roots of the quadratic t^2 - 12t + 32 = 0. Solving gives t = 4 and t = 8, so x and y can be 4 and 8 in either order. Then x^2 + y^2 = 4^2 + 8^2 = 16 + 64 = 80, which matches our result. This confirms that the identity method worked correctly and that the computed value is consistent.

Why Other Options Are Wrong:

• 24 is much smaller than the correct value and would correspond to a different set of x and y.
• 144 is equal to (x + y)^2 and ignores the subtraction of 2xy required by the identity.
• 128 could arise from incorrect manipulation, such as adding 2xy instead of subtracting.
• 76 is close but still wrong and likely reflects an arithmetic mistake when subtracting 2xy from 144.

Common Pitfalls:

• Forgetting the correct form of the identity and using x^2 + y^2 = (x + y)^2 + 2xy, which is incorrect.
• Making arithmetic errors while squaring 12 or computing 2 * 32.
• Trying to find x and y directly when it is not necessary, which may lead to algebra mistakes.

Final Answer:
80