Sure, let's solve this step by step.
Given the equation:
[tex]\[ x^2 + 2x + y^2 + 4y = 20 \][/tex]
We need to convert this into the standard form of a circle equation, which is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
To do this, we complete the square for both the [tex]\(x\)[/tex] terms and the [tex]\(y\)[/tex] terms.
1. Completing the square for the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 + 2x \][/tex]
We take the coefficient of [tex]\(x\)[/tex], which is 2, divide it by 2, and then square it:
[tex]\[ \left(\frac{2}{2}\right)^2 = 1 \][/tex]
So, we rewrite [tex]\( x^2 + 2x \)[/tex] as:
[tex]\[ (x + 1)^2 - 1 \][/tex]
2. Completing the square for the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 + 4y \][/tex]
We take the coefficient of [tex]\(y\)[/tex], which is 4, divide it by 2, and then square it:
[tex]\[ \left(\frac{4}{2}\right)^2 = 4 \][/tex]
So, we rewrite [tex]\( y^2 + 4y \)[/tex] as:
[tex]\[ (y + 2)^2 - 4 \][/tex]
3. Substitute these back into the original equation:
[tex]\[ (x + 1)^2 - 1 + (y + 2)^2 - 4 = 20 \][/tex]
Combine the constants on the left side of the equation:
[tex]\[
(x + 1)^2 + (y + 2)^2 - 5 = 20
\][/tex]
Add 5 to both sides to balance the equation:
[tex]\[ (x + 1)^2 + (y + 2)^2 = 25 \][/tex]
Now, we have the standard form of the circle equation:
[tex]\[ (x - (-1))^2 + (y - (-2))^2 = 5^2 \][/tex]
From this, we can identify the center [tex]\((h, k)\)[/tex] and the radius [tex]\(r\)[/tex]:
- The center, [tex]\((h, k)\)[/tex], is [tex]\((-1, -2)\)[/tex].
- The radius, [tex]\(r\)[/tex], is [tex]\(\sqrt{25} = 5\)[/tex].
Thus, the center of the circle is [tex]\((-1, -2)\)[/tex] and the radius is 5. Therefore, the correct answer is:
[tex]\[ \text{The center is located at } (-1, -2) \text{, and the radius is } 5 \text{.} \][/tex]
