We are tasked with finding the largest possible value of $x$ satisfying the equation

$x^2 + y^2 = 4x - 10y + 25.$

Step 1: Rewrite the equation

Rearrange all terms to one side: $x^2 - 4x + y^2 + 10y - 25 = 0.$

Step 2: Complete the square

For $x$-terms:

The terms involving $x$ are $x^2 - 4x$. Complete the square: $x^2 - 4x = (x - 2)^2 - 4.$

For $y$-terms:

The terms involving $y$ are $y^2 + 10y$. Complete the square: $y^2 + 10y = (y + 5)^2 - 25.$

Substitute these into the equation: $(x - 2)^2 - 4 + (y + 5)^2 - 25 - 25 = 0.$

Simplify: $(x - 2)^2 + (y + 5)^2 - 54 = 0.$

Rearrange: $(x - 2)^2 + (y + 5)^2 = 54.$

Step 3: Interpret geometrically

The equation represents a circle centered at $(2, -5)$ with radius $\sqrt{54} = 3\sqrt{6}$.

Step 4: Maximize $x$

The largest $x$-coordinate of any point on the circle is at the far-right edge of the circle, which occurs when we move the radius $3\sqrt{6}$ to the right of the center. The $x$-coordinate is: $x = 2 + 3\sqrt{6}.$

Final Answer:

$\boxed{2 + 3\sqrt{6}}$

This is the largest possible value of $x$.

Would you like a more detailed geometric explanation or further insights?

Related Questions:

1. How do we find the smallest value of $x$ in this problem?
2. What is the corresponding $y$-coordinate for the largest $x$?
3. Can the equation be rewritten as a parametric equation in terms of $t$?
4. How does completing the square help in solving geometric problems like this?
5. How can the general form of a circle be derived from such an equation?

Tip:

For optimization problems involving circles, visualizing the geometry is often the fastest approach!