Question

Complete the square to rewrite the following equation. Identify the center and radius of the circle. You must show all work and calculations to receive credit. (10 points)

x2 − 4x + y2 + 8y = −4

Answer 1

To complete the square and rewrite the given equation of the circle, we start with the equation:

\[
x^2 - 4x + y^2 + 8y = -4
\]

We want to reorganize and complete the square for the terms involving \(x\) and \(y\).

### Step 1: Complete the square for \(x\)

Take the \(x\) terms: \(x^2 - 4x\).

To complete the square:
1. Take the coefficient of \(x\) (which is \(-4\)), divide it by 2 to get \(-2\), and then square it: \((-2)^2 = 4\).
2. Add and subtract \(4\) inside the equation.

Thus, we have:
\[
x^2 - 4x = (x - 2)^2 - 4
\]

### Step 2: Complete the square for \(y\)

Next, take the \(y\) terms: \(y^2 + 8y\).

To complete the square:
1. Take the coefficient of \(y\) (which is \(8\)), divide it by 2 to get \(4\), and then square it: \(4^2 = 16\).
2. Add and subtract \(16\) inside the equation.

Thus, we have:
\[
y^2 + 8y = (y + 4)^2 - 16
\]

### Step 3: Substitute back into the equation

Now we substitute both completed squares back into the original equation:

\[
(x - 2)^2 - 4 + (y + 4)^2 - 16 = -4
\]

Combining the constant terms on the left side gives:

\[
(x - 2)^2 + (y + 4)^2 - 20 = -4
\]

Add \(20\) to both sides to isolate the completed squares:

\[
(x - 2)^2 + (y + 4)^2 = 16
\]

### Step 4: Identify the center and radius

This is now in the standard form of a circle's equation \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

From our equation \((x - 2)^2 + (y + 4)^2 = 16\):
- The center \((h, k)\) is \((2, -4)\).
- The radius \(r\) is given by \(r^2 = 16\), thus \(r = \sqrt{16} = 4\).

### Final Results

- **Center**: \((2, -4)\)
- **Radius**: \(4\)

The completed square form of the equation is:

\[
(x - 2)^2 + (y + 4)^2 = 16
\]