https://file.aiquickdraw.com/mathbot/file/1724754365087auedccg6.jpg

The given equation is in the general form:

\[
x^2 + y^2 + 14x + 36 = 0
\]

To find the center and radius of the circle, we need to convert this into the standard form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

### Steps to Convert:

1. **Rearrange the equation**:
   \[
   x^2 + 14x + y^2 = -36
   \]

2. **Complete the square for the \(x\) terms**:
   - Take the coefficient of \(x\), which is 14, divide it by 2, and square it: 
   \[
   \left(\frac{14}{2}\right)^2 = 49
   \]
   - Add and subtract 49 to complete the square:
   \[
   (x^2 + 14x + 49) + y^2 = -36 + 49
   \]

3. **Rewrite as a perfect square**:
   \[
   (x + 7)^2 + y^2 = 13
   \]

Now the equation is in standard form:
\[
(x + 7)^2 + y^2 = 13
\]

### Determine the Center and Radius:
- **Center** \((h, k)\): \((-7, 0)\)
- **Radius** \(r\): \(\sqrt{13}\)

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

Here are some related questions you might find helpful:
1. How do you derive the radius of a circle from the general form?
2. What are the steps for completing the square for both \(x\) and \(y\) terms if necessary?
3. How can you graph a circle using its center and radius?
4. How does the equation change if the circle is shifted vertically or horizontally?
5. What if the radius is zero—what does that imply about the circle?

**Tip**: When completing the square, remember to balance the equation by adding the same value to both sides after completing the square for each variable.

Learn how to convert a circle's equation from general form to standard form to find its center and radius. This detailed guide covers completing the square for both x and y terms, essential for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation