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The given equation is:

\[
4y^2 + x - 36y + 68 = 0
\]

### Step 1: Rearrange into Conic Form
To write this equation in standard conic form, we'll complete the square for the \( y \)-terms.

1. Group the terms involving \( y \):
   \[
   4y^2 - 36y + x + 68 = 0
   \]
   Separate the \( x \) and constant terms:
   \[
   4(y^2 - 9y) + x + 68 = 0
   \]

2. Complete the square for \( y^2 - 9y \). First, take half of 9, square it to get \( \left(\frac{9}{2}\right)^2 = \frac{81}{4} \), and add/subtract inside the parentheses.
   \[
   y^2 - 9y = (y - \frac{9}{2})^2 - \frac{81}{4}
   \]

3. Substitute this into the equation:
   \[
   4 \left( (y - \frac{9}{2})^2 - \frac{81}{4} \right) + x + 68 = 0
   \]
   Simplify:
   \[
   4(y - \frac{9}{2})^2 - 81 + x + 68 = 0
   \]
   \[
   4(y - \frac{9}{2})^2 + x - 13 = 0
   \]
   Finally, isolate the constants on one side:
   \[
   4(y - \frac{9}{2})^2 + x = 13
   \]

### Step 2: Identify the Conic Type
This equation is of the form:
\[
A(y - k)^2 + x = C
\]
This is the equation of a **parabola** that opens horizontally. 

### Summary:
The given equation represents a **parabola**. In standard form, it is written as:
\[
4(y - \frac{9}{2})^2 + x = 13
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you determine the direction of opening for a parabola?
2. What is the vertex of this parabola?
3. How do you convert a general conic equation to standard form?
4. What are the key features of a conic section?
5. How does completing the square help in identifying conic sections?

### Tip:
When completing the square, always remember to balance the equation by adjusting the constant terms on both sides!

4y^2 + x - 36y + 68 = 0. What type of conic? Write the equation in conic form.

Learn how to convert the equation 4y^2 + x - 36y + 68 = 0 into standard conic form by completing the square. Discover how to identify the type of conic section, in this case, a parabola, using step-by-step methods. Suitable for high school students.

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