https://file.aiquickdraw.com/mathbot/file/1728946651223dggis7as.jpg

The equation given is:

\[
x^2 + y^2 - 6x - 14y = -54
\]

This is the equation of a circle, but it's not in the standard form yet. To convert it to standard form, we need to complete the square for both \(x\) and \(y\).

### Step 1: Group the \(x\) terms and the \(y\) terms
\[
(x^2 - 6x) + (y^2 - 14y) = -54
\]

### Step 2: Complete the square for \(x\) and \(y\)

For \(x^2 - 6x\), take half of the coefficient of \(x\) (which is \(-6\)), square it, and add it inside the parentheses:
\[
\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9
\]
So, \(x^2 - 6x\) becomes \((x - 3)^2 - 9\).

For \(y^2 - 14y\), take half of the coefficient of \(y\) (which is \(-14\)), square it, and add it inside the parentheses:
\[
\left(\frac{-14}{2}\right)^2 = (-7)^2 = 49
\]
So, \(y^2 - 14y\) becomes \((y - 7)^2 - 49\).

Now, add \(9\) and \(49\) to both sides to maintain equality:
\[
(x - 3)^2 - 9 + (y - 7)^2 - 49 = -54
\]
\[
(x - 3)^2 + (y - 7)^2 - 58 = -54
\]
\[
(x - 3)^2 + (y - 7)^2 = 4
\]

### Step 3: Write the equation in standard form
The equation is now in the standard form of a circle:
\[
(x - 3)^2 + (y - 7)^2 = 4
\]

### Interpretation
- The center of the circle is at \((3, 7)\).
- The radius of the circle is \(\sqrt{4} = 2\).

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

Here are five related questions:
1. How do you find the radius of a circle from its equation?
2. What is the significance of completing the square in this process?
3. Can the center of a circle ever be outside the coordinate plane?
4. How would you graph a circle given its standard form equation?
5. How does the standard form equation differ from the general form?

**Tip**: Always remember to complete the square when converting a circle's general equation to standard form!

Consider the following equation of a circle: x^2 + y^2 - 6x - 14y = -54. Graph the circle.

Learn how to graph a circle by converting the equation x^2 + y^2 - 6x - 14y = -54 into standard form through completing the square. This step-by-step solution explains how to find the center and radius of the circle. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation