### Part A: Completing the Square
The initial equation given is:
[tex]\[ x^2 - 4x + y^2 + 8y = -4 \][/tex]
1. Group the x and y terms together:
[tex]\[ (x^2 - 4x) + (y^2 + 8y) = -4 \][/tex]
2. Complete the square for the x terms:
The expression [tex]\( x^2 - 4x \)[/tex] can be rewritten by completing the square.
- Take the coefficient of x, which is -4, divide it by 2, and square it:
[tex]\[ \left(-\frac{4}{2}\right)^2 = (-2)^2 = 4 \][/tex]
- Add and subtract this square inside the equation:
[tex]\[ (x^2 - 4x + 4 - 4) = (x - 2)^2 - 4 \][/tex]
3. Complete the square for the y terms:
The expression [tex]\( y^2 + 8y \)[/tex] can be rewritten by completing the square.
- Take the coefficient of y, which is 8, divide it by 2, and square it:
[tex]\[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \][/tex]
- Add and subtract this square inside the equation:
[tex]\[ (y^2 + 8y + 16 - 16) = (y + 4)^2 - 16 \][/tex]
4. Rewrite the equation with the completed squares:
[tex]\[ (x - 2)^2 - 4 + (y + 4)^2 - 16 = -4 \][/tex]
5. Move the constant terms to the right side:
Combine the constants -4 and -16 on the left side, and simplify the right side:
[tex]\[ (x - 2)^2 + (y + 4)^2 - 20 = -4 \][/tex]
[tex]\[ (x - 2)^2 + (y + 4)^2 = -4 + 20 \][/tex]
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
So, the equation in standard form is:
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
### Part B: Center and Radius of the Circle
The standard form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
By comparing this with our derived equation:
[tex]\[ (x - 2)^2 + (y + 4)^2 = 16 \][/tex]
- We can identify the center [tex]\((h, k)\)[/tex]:
[tex]\[ h = 2 \][/tex]
[tex]\[ k = -4 \][/tex]
So the center is [tex]\((2, -4)\)[/tex].
- The radius [tex]\(r\)[/tex] is found by taking the square root of the constant term on the right side:
[tex]\[ r^2 = 16 \][/tex]
[tex]\[ r = \sqrt{16} = 4 \][/tex]
Thus, the center and radius of the circle are:
- Center: [tex]\((2, -4)\)[/tex]
- Radius: [tex]\(4\)[/tex]
