### Part A: Completing the Square
We start with the given equation:
[tex]\[ x^2 + 2x + y^2 + 4y = 20 \][/tex]
To rewrite this equation in standard form, we need to complete the square for both the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
#### Step 1: Completing the Square for [tex]\( x \)[/tex]
First, consider the [tex]\( x \)[/tex]-terms:
[tex]\[ x^2 + 2x \][/tex]
To complete the square, we:
1. Take the coefficient of [tex]\( x \)[/tex], which is [tex]\( 2 \)[/tex],
2. Divide it by 2 to get [tex]\( 1 \)[/tex],
3. Square it to get [tex]\( 1 \)[/tex].
Now, rewrite [tex]\( x^2 + 2x \)[/tex]:
[tex]\[ x^2 + 2x = (x + 1)^2 - 1 \][/tex]
So, [tex]\( x^2 + 2x \)[/tex] becomes [tex]\( (x + 1)^2 - 1 \)[/tex].
#### Step 2: Completing the Square for [tex]\( y \)[/tex]
Now, consider the [tex]\( y \)[/tex]-terms:
[tex]\[ y^2 + 4y \][/tex]
To complete the square, we:
1. Take the coefficient of [tex]\( y \)[/tex], which is [tex]\( 4 \)[/tex],
2. Divide it by 2 to get [tex]\( 2 \)[/tex],
3. Square it to get [tex]\( 4 \)[/tex].
Now, rewrite [tex]\( y^2 + 4y \)[/tex]:
[tex]\[ y^2 + 4y = (y + 2)^2 - 4 \][/tex]
So, [tex]\( y^2 + 4y \)[/tex] becomes [tex]\( (y + 2)^2 - 4 \)[/tex].
#### Step 3: Substitute Back into the Equation
Now substitute these completed squares back into the original equation:
[tex]\[ x^2 + 2x + y^2 + 4y = (x + 1)^2 - 1 + (y + 2)^2 - 4 \][/tex]
Simplify and combine like terms:
[tex]\[
(x + 1)^2 - 1 + (y + 2)^2 - 4 = 20 \\
(x + 1)^2 + (y + 2)^2 - 5 = 20 \\
(x + 1)^2 + (y + 2)^2 = 25
\][/tex]
So, the standard form of the equation is:
[tex]\[ (x + 1)^2 + (y + 2)^2 = 25 \][/tex]
### Part B: Finding the Center and Radius
The standard form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Comparing this with our standard form:
[tex]\[ (x + 1)^2 + (y + 2)^2 = 25 \][/tex]
We can see that:
- [tex]\( h = -1 \)[/tex] (because [tex]\( x + 1 \)[/tex] is the same as [tex]\( x - (-1) \)[/tex])
- [tex]\( k = -2 \)[/tex] (because [tex]\( y + 2 \)[/tex] is the same as [tex]\( y - (-2) \)[/tex])
- [tex]\( r^2 = 25 \)[/tex]
So, the center (h, k) of the circle is:
[tex]\[ (-1, -2) \][/tex]
And the radius [tex]\( r \)[/tex] is:
[tex]\[ r = \sqrt{25} = 5 \][/tex]
### Summary
- Standard Form: [tex]\[ (x + 1)^2 + (y + 2)^2 = 25 \][/tex]
- Center: [tex]\[ (-1, -2) \][/tex]
- Radius: [tex]\[ 5 \][/tex]
