1Rewrite the given equation in standard form for an ellipse
36
𝑥
2
+
20
𝑦
2
−
144
𝑥
+
120
𝑦
−
396
=
0
36x 
2
 +20y 
2
 −144x+120y−396=0 with step by step solution 

To rewrite the given equation in the standard form for an ellipse, we will follow these steps:

1. **Rewrite the equation** by grouping the \(x\)-terms and \(y\)-terms together.
2. **Complete the square** for both the \(x\)-terms and the \(y\)-terms.
3. **Divide through by the constant** to set the equation equal to 1, which is the standard form for an ellipse.

Let's go through each step in detail.

### Step 1: Rewrite the Equation

Given equation:
\[
36x^2 + 20y^2 - 144x + 120y - 396 = 0
\]

Group the \(x\)-terms and \(y\)-terms:
\[
36x^2 - 144x + 20y^2 + 120y = 396
\]

### Step 2: Complete the Square

**For the \(x\)-terms:**

Factor out the coefficient of \(x^2\) from the \(x\)-terms:
\[
36(x^2 - 4x) + 20y^2 + 120y = 396
\]

To complete the square, take half of the coefficient of \(x\) (which is \(-4\)), square it, and add it inside the parentheses:
\[
\text{Half of } -4 \text{ is } -2, \text{ and } (-2)^2 = 4.
\]

Add and subtract 4 inside the parentheses:
\[
36(x^2 - 4x + 4 - 4) + 20y^2 + 120y = 396
\]

Rewrite:
\[
36((x - 2)^2 - 4) + 20y^2 + 120y = 396
\]

Distribute the 36:
\[
36(x - 2)^2 - 144 + 20y^2 + 120y = 396
\]

**For the \(y\)-terms:**

Factor out the coefficient of \(y^2\) from the \(y\)-terms:
\[
36(x - 2)^2 - 144 + 20(y^2 + 6y) = 396
\]

To complete the square, take half of the coefficient of \(y\) (which is \(6\)), square it, and add it inside the parentheses:
\[
\text{Half of } 6 \text{ is } 3, \text{ and } (3)^2 = 9.
\]

Add and subtract 9 inside the parentheses:
\[
36(x - 2)^2 - 144 + 20(y^2 + 6y + 9 - 9) = 396
\]

Rewrite:
\[
36(x - 2)^2 - 144 + 20((y + 3)^2 - 9) = 396
\]

Distribute the 20:
\[
36(x - 2)^2 - 144 + 20(y + 3)^2 - 180 = 396
\]

### Step 3: Simplify and Divide Through by the Constant

Combine like terms:
\[
36(x - 2)^2 + 20(y + 3)^2 - 324 = 396
\]

Add 324 to both sides:
\[
36(x - 2)^2 + 20(y + 3)^2 = 720
\]

Divide the entire equation by 720 to set it equal to 1:
\[
\frac{36(x - 2)^2}{720} + \frac{20(y + 3)^2}{720} = \frac{720}{720}
\]

Simplify each term:
\[
\frac{(x - 2)^2}{20} + \frac{(y + 3)^2}{36} = 1
\]

### Final Standard Form of the Ellipse

\[
\frac{(x - 2)^2}{20} + \frac{(y + 3)^2}{36} = 1
\]

This is the standard form of an ellipse centered at \((2, -3)\) with semi-major axis \(\sqrt{36} = 6\) and semi-minor axis \(\sqrt{20} = 2\sqrt{5}\).

Would you like to ask any questions or see more details about any of the steps? Here are some related questions you might find interesting:

1. How do you determine the center of an ellipse from its standard form?
2. What is the difference between the major and minor axes of an ellipse?
3. How can you convert an ellipse equation from general form to standard form?
4. What are the steps for completing the square for a quadratic expression?
5. How do you find the lengths of the semi-major and semi-minor axes of an ellipse?

**Tip:** Always remember to factor out the coefficient of the squared term when completing the square in an equation.

Learn how to rewrite the equation of an ellipse in standard form with a detailed step-by-step solution. This problem involves completing the square for both x and y terms, converting a general form equation to standard form, and understanding ellipse properties. Suitable for high school students.

Math Problem Statement

1Rewrite the given equation in standard form for an ellipse 36 𝑥 2 + 20 𝑦 2 − 144 𝑥 + 120 𝑦 − 396

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