To find the standard form of the given ellipse, we need to rewrite the equation:

\[
25x^2 + 9y^2 - 100x - 54y - 44 = 0
\]

### Step 1: Group and Rearrange Terms
Group the \(x\) and \(y\) terms together:

\[
25x^2 - 100x + 9y^2 - 54y = 44
\]

### Step 2: Complete the Square for \(x\) and \(y\)
First, factor out the coefficients of \(x^2\) and \(y^2\):

\[
25(x^2 - 4x) + 9(y^2 - 6y) = 44
\]

Now, complete the square for both \(x\) and \(y\).

- For \(x\), take half of \(-4\), square it, and add it inside the parentheses:

\[
x^2 - 4x \rightarrow (x - 2)^2 - 4
\]

- For \(y\), take half of \(-6\), square it, and add it inside the parentheses:

\[
y^2 - 6y \rightarrow (y - 3)^2 - 9
\]

Substitute these back into the equation:

\[
25((x - 2)^2 - 4) + 9((y - 3)^2 - 9) = 44
\]

Distribute and simplify:

\[
25(x - 2)^2 - 100 + 9(y - 3)^2 - 81 = 44
\]

Combine like terms:

\[
25(x - 2)^2 + 9(y - 3)^2 = 225
\]

### Step 3: Divide by 225 to Get the Standard Form
Divide the entire equation by 225:

\[
\frac{25(x - 2)^2}{225} + \frac{9(y - 3)^2}{225} = 1
\]

Simplify the fractions:

\[
\frac{(x - 2)^2}{9} + \frac{(y - 3)^2}{25} = 1
\]

### Step 4: Identify the Properties of the Ellipse
The equation is now in standard form:

\[
\frac{(x - 2)^2}{3^2} + \frac{(y - 3)^2}{5^2} = 1
\]

From this, we can see:

- The center of the ellipse is at \((2, 3)\).
- The semi-major axis is along the \(y\)-axis with a length of 5.
- The semi-minor axis is along the \(x\)-axis with a length of 3.

Would you like a deeper explanation, or do you have any specific questions about this process?

### Follow-Up Questions

1. How do we determine the orientation of the ellipse's major and minor axes?
2. What is the significance of completing the square when working with conic sections?
3. How can we find the foci of the ellipse given in standard form?
4. Can you provide an example of finding the equation of a tangent line to the ellipse at a specific point?
5. How would the equation change if the ellipse were centered at a different point?

### Tip
When completing the square, always ensure to balance the equation by adding or subtracting the same values on both sides.

ellipse 25x^2 + 9y^2 - 100x - 54y - 44 = 0

Explore the process of converting the equation 25x^2 + 9y^2 - 100x - 54y - 44 = 0 into the standard form of an ellipse. This detailed step-by-step solution covers key concepts such as completing the square and identifying ellipse properties. Suitable for high school students in Grades 10-12.

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