We are asked to find $9x^2 + 4y^2$, given two conditions: $xy = 6$ and $3x + 2y = 12$.

Step 1: Solve for $y$ in terms of $x$

From the equation $3x + 2y = 12$, we can isolate $y$: $2y = 12 - 3x$ $y = \frac{12 - 3x}{2}$

Step 2: Substitute $y$ in the equation $xy = 6$

Now substitute this expression for $y$ into the equation $xy = 6$: $x \left( \frac{12 - 3x}{2} \right) = 6$ Multiplying through by 2 to eliminate the denominator: $x(12 - 3x) = 12$ Expanding: $12x - 3x^2 = 12$ Rearranging: $3x^2 - 12x + 12 = 0$ Dividing the entire equation by 3: $x^2 - 4x + 4 = 0$ Factoring the quadratic: $(x - 2)^2 = 0$ Thus, $x = 2$.

Step 3: Solve for $y$

Now substitute $x = 2$ into the equation $3x + 2y = 12$: $3(2) + 2y = 12$ $6 + 2y = 12$ $2y = 6 \quad \Rightarrow \quad y = 3$

Step 4: Calculate $9x^2 + 4y^2$

Now that we know $x = 2$ and $y = 3$, we can calculate $9x^2 + 4y^2$: $9x^2 + 4y^2 = 9(2^2) + 4(3^2)$ $= 9(4) + 4(9)$ $= 36 + 36 = 72$

Thus, the value of $9x^2 + 4y^2$ is $\boxed{72}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the method to factor quadratic equations like $x^2 - 4x + 4$?
2. How can we solve a system of equations with substitution?
3. What are the properties of perfect square trinomials?
4. How does completing the square differ from factoring?
5. How do we graph the equation $3x + 2y = 12$?

Tip: When solving a system of equations, substitution is useful when one equation can be easily solved for one variable in terms of the other.