Given 3x + 2y = 12 and xy = 6 in basic algebra, compute the exact value of 9x^2 + 4y^2. Use identities rather than solving for x and y explicitly.

Introduction / Context:
This problem checks fluency with algebraic identities that allow direct evaluation of expressions without first solving for the individual variables. Recognizing the pattern behind (3x + 2y)^2 helps collapse the computation quickly and accurately.

Given Data / Assumptions:

• 3x + 2y = 12
• xy = 6
• We must find 9x^2 + 4y^2, which equals (3x)^2 + (2y)^2.

Concept / Approach:
Use the expansion (A + B)^2 = A^2 + B^2 + 2AB with A = 3x and B = 2y. Then (3x + 2y)^2 = 9x^2 + 4y^2 + 12xy. Since both (3x + 2y) and xy are known, we can isolate 9x^2 + 4y^2 directly.

Step-by-Step Solution:
Compute (3x + 2y)^2 = 12^2 = 144.From identity: (3x + 2y)^2 = 9x^2 + 4y^2 + 12xy.Hence 9x^2 + 4y^2 = 144 − 12xy.Substitute xy = 6: 9x^2 + 4y^2 = 144 − 12*6 = 144 − 72 = 72.

Verification / Alternative check:
If desired, solve for one variable and check numerically. For instance, set y = (12 − 3x)/2, substitute into xy = 6, solve, and confirm that 9x^2 + 4y^2 evaluates to 72 for both roots. The identity method is faster and avoids quadratic arithmetic.

Why Other Options Are Wrong:

• 80, 76, 74, 68: These come from arithmetic slips such as squaring 12 incorrectly, forgetting the 12xy term, or using 2xy instead of 12xy.

Common Pitfalls:
Expanding (3x + 2y)^2 but omitting the middle term; trying to solve for x and y first, which is unnecessary overhead and error-prone.

Final Answer:
72