The problem asks us to determine the radius of the circle given the equation:

$x^2 + y^2 - 8x + 6y + c = 0,$

where $c = -24$.

Step 1: Rewrite the equation

Substitute $c = -24$ into the equation: $x^2 + y^2 - 8x + 6y - 24 = 0.$

Rearrange the terms: $x^2 - 8x + y^2 + 6y = 24.$

Step 2: Complete the square

For the $x$-terms:

Take $x^2 - 8x$ and complete the square:

• The coefficient of $x$ is $-8$. Half of it is $-4$, and squaring it gives $16$.
• Add and subtract $16$: $x^2 - 8x = (x - 4)^2 - 16.$

For the $y$-terms:

Take $y^2 + 6y$ and complete the square:

• The coefficient of $y$ is $6$. Half of it is $3$, and squaring it gives $9$.
• Add and subtract $9$: $y^2 + 6y = (y + 3)^2 - 9.$

Step 3: Rewrite the equation in standard form

Substitute the completed squares into the equation: $(x - 4)^2 - 16 + (y + 3)^2 - 9 = 24.$

Simplify: $(x - 4)^2 + (y + 3)^2 - 25 = 24.$

Add $25$ to both sides: $(x - 4)^2 + (y + 3)^2 = 49.$

This is the standard form of the equation of a circle: $(x - h)^2 + (y - k)^2 = r^2,$ where $(h, k) = (4, -3)$ is the center and $r^2 = 49$.

Step 4: Solve for the radius

$r = \sqrt{49} = 7.$

Final Answer:

The radius of the circle is: $\boxed{7.00}.$

Would you like more details or a further explanation?

Here are some related questions:

1. What is the process of completing the square?
2. How can you derive the equation of a circle in standard form from general form?
3. Can you graph a circle given its standard form equation?
4. How do changes in the value of $c$ affect the circle's radius?
5. What is the significance of the circle’s center in geometry?

Tip: Completing the square is a powerful tool for rewriting quadratic expressions in vertex or standard form.