To convert the equation $4x^2 + 9y^2 - 36x + 18y + 36 = 0$ to its standard form, we will complete the square for both the $x$ and $y$ terms.

Step 1: Group the $x$ and $y$ terms together

$4x^2 - 36x + 9y^2 + 18y = -36$

Step 2: Factor out the coefficients of $x^2$ and $y^2$ from the grouped terms

$4(x^2 - 9x) + 9(y^2 + 2y) = -36$

Step 3: Complete the square for both $x$ and $y$

• For $x$, complete the square inside the parentheses: $x^2 - 9x \quad \text{can be written as} \quad (x - \frac{9}{2})^2 - \left(\frac{9}{2}\right)^2$ So, $x^2 - 9x = (x - \frac{9}{2})^2 - \frac{81}{4}$
• For $y$, complete the square inside the parentheses: $y^2 + 2y \quad \text{can be written as} \quad (y + 1)^2 - 1$ So, $y^2 + 2y = (y + 1)^2 - 1$

Step 4: Substitute the completed squares back into the equation

[ 4\left[(x - \frac{9}{2})^2 - \frac{81}{4}\right] + 9\left[(y + 1)^2 - 1\right] = -36 ] Expand and simplify: $4(x - \frac{9}{2})^2 - 81 + 9(y + 1)^2 - 9 = -36$ $4(x - \frac{9}{2})^2 + 9(y + 1)^2 - 90 = -36$

Step 5: Move constants to the other side

$4(x - \frac{9}{2})^2 + 9(y + 1)^2 = 54$

Step 6: Divide by 54 to normalize the equation

$\frac{4(x - \frac{9}{2})^2}{54} + \frac{9(y + 1)^2}{54} = 1$

Simplify the coefficients: $\frac{(x - \frac{9}{2})^2}{\frac{27}{2}} + \frac{(y + 1)^2}{6} = 1$ Which simplifies to: $\frac{(x - \frac{9}{2})^2}{\frac{27}{2}} + \frac{(y + 1)^2}{6} = 1$

Thus, the equation in standard form is: $\frac{(x - \frac{9}{2})^2}{\frac{27}{2}} + \frac{(y + 1)^2}{6} = 1$

This is the standard form of the ellipse.

Would you like more details or have any questions about this process?

Here are some related questions you might consider:

1. What are the center, axes, and orientation of the ellipse?
2. How do you determine the lengths of the major and minor axes?
3. How does completing the square work in general?
4. What is the geometric interpretation of this ellipse?
5. How would this process change if the original equation were for a hyperbola?

Tip: Completing the square is a key technique in transforming quadratic equations into standard forms. It's especially useful in conic sections.