To solve the given equation:

\[
25y^2 - 36x^2 + 150x - 216y = 1296
\]

we can start by identifying the type of equation it represents and try to simplify it. The equation appears to be a conic section, which could represent an ellipse, hyperbola, or parabola, depending on how the terms are organized. To proceed, we will attempt to rewrite it in a standard form by completing the square for both \( x \) and \( y \).

### Step 1: Group the \( x \) and \( y \) terms together
Rearrange the equation to group the \( x \) terms together and the \( y \) terms together:

\[
-36x^2 + 150x + 25y^2 - 216y = 1296
\]

### Step 2: Factor out the coefficients of \( x^2 \) and \( y^2 \)
Factor out the coefficients of the squared terms:

\[
-36(x^2 - \frac{25}{12}x) + 25(y^2 - \frac{216}{25}y) = 1296
\]

### Step 3: Complete the square for both \( x \) and \( y \)

#### For \( x \):

To complete the square for \( x \), take half of the coefficient of \( x \) (which is \(-\frac{25}{12}\)), square it, and add/subtract inside the parentheses:

\[
x^2 - \frac{25}{12}x = \left(x - \frac{25}{24}\right)^2 - \left(\frac{25}{24}\right)^2
\]

#### For \( y \):

To complete the square for \( y \), take half of the coefficient of \( y \) (which is \(-\frac{216}{25}\)), square it, and add/subtract inside the parentheses:

\[
y^2 - \frac{216}{25}y = \left(y - \frac{108}{25}\right)^2 - \left(\frac{108}{25}\right)^2
\]

Now, substitute these completed squares back into the equation:

\[
-36 \left[ \left(x - \frac{25}{24}\right)^2 - \left(\frac{25}{24}\right)^2 \right] + 25 \left[ \left(y - \frac{108}{25}\right)^2 - \left(\frac{108}{25}\right)^2 \right] = 1296
\]

### Step 4: Distribute and simplify

Expand and rearrange:

\[
-36 \left(x - \frac{25}{24}\right)^2 + 36 \left(\frac{25}{24}\right)^2 + 25 \left(y - \frac{108}{25}\right)^2 - 25 \left(\frac{108}{25}\right)^2 = 1296
\]

Combine the constants and move them to the right:

\[
-36 \left(x - \frac{25}{24}\right)^2 + 25 \left(y - \frac{108}{25}\right)^2 = 1296 - 36 \left(\frac{25}{24}\right)^2 + 25 \left(\frac{108}{25}\right)^2
\]

Calculate the constants:

\[
36 \left(\frac{25}{24}\right)^2 = \frac{375}{4}, \quad 25 \left(\frac{108}{25}\right)^2 = 11664
\]

Substitute these values:

\[
-36 \left(x - \frac{25}{24}\right)^2 + 25 \left(y - \frac{108}{25}\right)^2 = 1296 - \frac{375}{4} - 11664
\]

Simplify the right-hand side:

\[
1296 - \frac{375}{4} - 11664 = -12432.75
\]

### Step 5: Rewrite the equation in standard form

Divide the entire equation by \(-12432.75\) to get the standard form:

\[
\frac{\left(x - \frac{25}{24}\right)^2}{\frac{345.35}{12432.75}} - \frac{\left(y - \frac{108}{25}\right)^2}{\frac{500}{12432.75}} = 1
\]

### Conclusion

The equation represents a hyperbola.

Would you like more details or have any questions?

#### Related Questions:

1. How do we identify different types of conic sections?
2. What is the importance of completing the square in conic sections?
3. How to graph the hyperbola given its standard form?
4. How to find the center, foci, and vertices of the hyperbola?
5. What are the real-life applications of hyperbolas?

**Tip:** Completing the square helps in transforming any quadratic equation into its standard form, making it easier to analyze and graph.

Learn how to rewrite and solve the hyperbola equation 25y² - 36x² + 150x - 216y = 1296 by completing the square and simplifying to its standard form. This problem focuses on conic sections, algebra, and completing the square. Suitable for high school students in Grades 10-12.

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