2. Find the intercepts of the following algebraically, show your work. (check in your calculator if
you want some extra practice) You may do these on this sheet if you prefer. (x − 3)^2 + (y − 6)^2 = 58

To find the intercepts of the given equation \((x - 3)^2 + (y - 6)^2 = 58\), we'll solve for both the x-intercepts and y-intercepts separately.

### Finding the x-intercepts:
For x-intercepts, we set \(y = 0\) and solve for \(x\).

Substitute \(y = 0\) into the equation:
\[
(x - 3)^2 + (0 - 6)^2 = 58
\]

Simplify:
\[
(x - 3)^2 + 36 = 58
\]

Subtract 36 from both sides:
\[
(x - 3)^2 = 22
\]

Take the square root of both sides:
\[
x - 3 = \pm \sqrt{22}
\]

Solve for \(x\):
\[
x = 3 \pm \sqrt{22}
\]

Thus, the x-intercepts are:
\[
x = 3 + \sqrt{22} \quad \text{and} \quad x = 3 - \sqrt{22}
\]

### Finding the y-intercepts:
For y-intercepts, we set \(x = 0\) and solve for \(y\).

Substitute \(x = 0\) into the equation:
\[
(0 - 3)^2 + (y - 6)^2 = 58
\]

Simplify:
\[
9 + (y - 6)^2 = 58
\]

Subtract 9 from both sides:
\[
(y - 6)^2 = 49
\]

Take the square root of both sides:
\[
y - 6 = \pm 7
\]

Solve for \(y\):
\[
y = 6 + 7 \quad \text{or} \quad y = 6 - 7
\]

Thus, the y-intercepts are:
\[
y = 13 \quad \text{and} \quad y = -1
\]

### Summary of Intercepts:
- **x-intercepts:** \(x = 3 + \sqrt{22}\) and \(x = 3 - \sqrt{22}\)
- **y-intercepts:** \(y = 13\) and \(y = -1\)

Would you like any additional details or have any questions?

Here are some related questions for further practice:

1. Find the intercepts of \((x + 4)^2 + (y - 5)^2 = 29\).
2. Determine the x-intercepts of \(x^2 + y^2 = 25\).
3. Calculate the y-intercepts of \((x - 2)^2 + (y + 3)^2 = 50\).
4. Solve for the intercepts of the ellipse equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).
5. Find the intercepts for the hyperbola \(\frac{x^2}{16} - \frac{y^2}{25} = 1\).
6. Determine the intercepts of \((x - 1)^2 + (y + 2)^2 = 34\).
7. Calculate the x-intercepts of \(x^2 + y^2 = 49\).
8. Find the intercepts of the circle equation \((x + 3)^2 + (y - 7)^2 = 65\).

**Tip:** When dealing with intercepts, always set the opposite variable to zero and solve the resulting equation carefully.

Learn how to find the x-intercepts and y-intercepts of the circle equation (x - 3)^2 + (y - 6)^2 = 58. Follow step-by-step solutions for both intercept types, including calculations and formulas used. Suitable for high school students learning circle equations and intercepts.

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