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.

Find the value of 9x^2 + 4y^2 given xy=6 and 3x+2y=12 using algebraic methods. This step-by-step guide covers solving systems of equations, substitution, and quadratic equations. Suitable for students in Grades 8-10.

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