To solve the problem, we are tasked with finding the x-intercepts for the given equation of a circle:

$(x - 1)^2 + y^2 = 9$

Step 1: Recall the x-intercept condition

At the x-intercepts, the value of $y = 0$. Substitute $y = 0$ into the equation:

$(x - 1)^2 + 0^2 = 9$

This simplifies to: $(x - 1)^2 = 9$

Step 2: Solve for $x$

To solve $(x - 1)^2 = 9$, take the square root of both sides:

$x - 1 = \pm 3$

This gives two solutions: $x - 1 = 3 \quad \text{or} \quad x - 1 = -3$

Simplify both cases:

1. $x = 4$
2. $x = -2$

Step 3: Write the x-intercepts

The x-intercepts are: $(4, 0) \quad \text{and} \quad (-2, 0)$

Let me know if you have questions or need further clarification.

Here are 5 related questions to deepen understanding:

1. How do you find the y-intercepts for a circle equation?
2. What is the center and radius of the circle $(x - 1)^2 + y^2 = 9$?
3. How can you determine if a line intersects this circle?
4. What is the general equation for a circle in standard form?
5. How would the x-intercepts change if the radius were increased to 16?

Tip: Always substitute $y = 0$ (or $x = 0$) for finding intercepts efficiently!