To find the radius of the circle given by the equation [tex]\( x^2 + 12x + y^2 + 10y = -36 \)[/tex], we need to rewrite the equation in the standard form of a circle's equation, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here is the step-by-step process to convert the given equation to the standard form:
1. Group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 + 12x + y^2 + 10y = -36
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
- Take the coefficient of [tex]\(x\)[/tex] (which is 12), divide it by 2, and square the result:
[tex]\[
\left(\frac{12}{2}\right)^2 = 36
\][/tex]
- Add and subtract 36 inside the equation to complete the square:
[tex]\[
x^2 + 12x + 36 - 36
\][/tex]
- This can be rewritten as:
[tex]\[
(x + 6)^2 - 36
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
- Take the coefficient of [tex]\(y\)[/tex] (which is 10), divide it by 2, and square the result:
[tex]\[
\left(\frac{10}{2}\right)^2 = 25
\][/tex]
- Add and subtract 25 inside the equation to complete the square:
[tex]\[
y^2 + 10y + 25 - 25
\][/tex]
- This can be rewritten as:
[tex]\[
(y + 5)^2 - 25
\][/tex]
4. Substitute these completed squares into the original equation:
[tex]\[
(x + 6)^2 - 36 + (y + 5)^2 - 25 = -36
\][/tex]
5. Combine like terms:
[tex]\[
(x + 6)^2 + (y + 5)^2 - 61 = -36
\][/tex]
[tex]\[
(x + 6)^2 + (y + 5)^2 = -36 + 61
\][/tex]
[tex]\[
(x + 6)^2 + (y + 5)^2 = 25
\][/tex]
6. Compare this result with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]:
- Here [tex]\(r^2 = 25\)[/tex], therefore, [tex]\(r = \sqrt{25} = 5\)[/tex]
So, the radius of the circle is:
[tex]\[
\boxed{5}
\][/tex]
