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Let's work through the conversion of the given equation from general form to standard form step by step. The general form of the equation is:

\[
9x^2 + 4y^2 + 72x - 48y + 144 = 0
\]

### Step 1: Group the \(x\)-terms and the \(y\)-terms.
We group the quadratic and linear terms involving \(x\) and \(y\):

\[
9(x^2 + 8x) + 4(y^2 - 12y) = -144
\]

### Step 2: Complete the square.
Now, we complete the square for both the \(x\)-terms and the \(y\)-terms.

#### For the \(x\)-terms:
The expression is \(x^2 + 8x\). To complete the square, take half of 8, which is 4, and then square it, giving 16. So:

\[
x^2 + 8x = (x + 4)^2 - 16
\]

#### For the \(y\)-terms:
The expression is \(y^2 - 12y\). Take half of -12, which is -6, and square it to get 36. So:

\[
y^2 - 12y = (y - 6)^2 - 36
\]

Now substitute these into the equation:

\[
9((x + 4)^2 - 16) + 4((y - 6)^2 - 36) = -144
\]

### Step 3: Simplify and move constants to the right-hand side.
Distribute the 9 and 4:

\[
9(x + 4)^2 - 144 + 4(y - 6)^2 - 144 = -144
\]

Combine like terms:

\[
9(x + 4)^2 + 4(y - 6)^2 - 288 = -144
\]

Add 288 to both sides:

\[
9(x + 4)^2 + 4(y - 6)^2 = 144
\]

### Step 4: Divide by 144 to get the equation in standard form.
Divide everything by 144:

\[
\frac{9(x + 4)^2}{144} + \frac{4(y - 6)^2}{144} = 1
\]

Simplify the fractions:

\[
\frac{(x + 4)^2}{16} + \frac{(y - 6)^2}{36} = 1
\]

### Step 5: Identify the values of \(h\), \(k\), \(D\), and \(E\).
From the standard form:

\[
\frac{(x - h)^2}{D} + \frac{(y - k)^2}{E} = 1
\]

we can compare with our result:

\[
\frac{(x + 4)^2}{16} + \frac{(y - 6)^2}{36} = 1
\]

Thus:
- \(h = -4\)
- \(k = 6\)
- \(D = 16\)
- \(E = 36\)

---

Would you like further details on any of the steps, or do you have any questions?

### Related Questions:
1. How do you complete the square in quadratic expressions?
2. What is the process for converting a general conic equation to standard form?
3. What do \(h\) and \(k\) represent in the standard form of an ellipse?
4. How can you determine if a given conic equation represents a circle, ellipse, or hyperbola?
5. Why do we divide both sides of the equation by the constant term to standardize the form?

### Tip:
Always double-check your completion of the square and constants when converting from general to standard form to avoid mistakes.

Convert general form to standard form: 9x^2 + 4y^2 + 72x - 48y + 144 = 0

Learn how to convert the general form of the equation 9x^2 + 4y^2 + 72x - 48y + 144 = 0 into its standard form. This problem focuses on key algebraic techniques like completing the square and identifying the parameters of an ellipse. Suitable for students in Grades 10-12.

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