Find the center and radius of the circle x^2+y^2-x=0

The given equation of the circle is:

\[
x^2 + y^2 - x = 0
\]

### Step 1: Rearrange the equation
We can rewrite the equation as:

\[
x^2 - x + y^2 = 0
\]

### Step 2: Complete the square
To find the center and radius, we need to complete the square for the \(x\)-terms.

Take the coefficient of \(x\), divide by 2, and square it. The coefficient of \(x\) is \(-1\), so:

\[
\left(\frac{-1}{2}\right)^2 = \frac{1}{4}
\]

Now, add and subtract \(\frac{1}{4}\) inside the equation:

\[
x^2 - x + \frac{1}{4} + y^2 = \frac{1}{4}
\]

This simplifies to:

\[
\left(x - \frac{1}{2}\right)^2 + y^2 = \frac{1}{4}
\]

### Step 3: Interpret the equation
This is now in the standard form of a circle equation:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center and \(r\) is the radius.

From the equation \(\left(x - \frac{1}{2}\right)^2 + y^2 = \frac{1}{4}\):
- The center is \(\left(\frac{1}{2}, 0\right)\)
- The radius is \(r = \frac{1}{2}\)

### Final Answer:
- **Center**: \(\left(\frac{1}{2}, 0\right)\)
- **Radius**: \(\frac{1}{2}\)

Would you like more details, or do you have any questions?

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Here are 5 related questions you might find helpful:
1. How do you complete the square for other equations involving circles?
2. What is the general form of a circle's equation in standard form?
3. How can you derive the equation of a circle given the center and radius?
4. What happens when a circle is shifted along the coordinate plane?
5. How does the distance formula relate to the equation of a circle?

**Tip**: Completing the square is a useful technique not just for circles but also for quadratic equations in general.

Find the center and radius of the circle x^2 + y^2 - x = 0

Learn how to find the center and radius of the circle given by the equation x^2 + y^2 - x = 0 using the completing the square method. This step-by-step guide covers important concepts in circle geometry and algebra. Suitable for high school students in Grades 9-12.

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