Question

Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work.
x2 + 2x + y2 + 4y = 20

Part B: What are the center and radius of the circle?

Answer 1

To rewrite the equation \(x^2 + 2x + y^2 + 4y = 20\) in standard form, we need to complete the square for both \(x\) and \(y\).

### Part A: Completing the Square

1. **Group the terms involving \(x\) and \(y\):**

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

2. **Complete the square for \(x\):**

- Take the coefficient of \(x\), which is \(2\), halve it to get \(1\), and square it to get \(1\).
- Add and subtract \(1\) inside the equation.

\[
x^2 + 2x + 1 - 1
\]

This can be rewritten as:

\[
(x + 1)^2 - 1
\]

3. **Complete the square for \(y\):**

- Take the coefficient of \(y\), which is \(4\), halve it to get \(2\), and square it to get \(4\).
- Add and subtract \(4\) inside the equation.

\[
y^2 + 4y + 4 - 4
\]

This can be rewritten as:

\[
(y + 2)^2 - 4
\]

4. **Substituting back into the equation:**

Now substitute both completed squares back into the equation:

\[
\big((x + 1)^2 - 1\big) + \big((y + 2)^2 - 4\big) = 20
\]

Simplifying gives:

\[
(x + 1)^2 + (y + 2)^2 - 1 - 4 = 20
\]

This simplifies to:

\[
(x + 1)^2 + (y + 2)^2 - 5 = 20
\]

Now, add \(5\) to both sides:

\[
(x + 1)^2 + (y + 2)^2 = 25
\]

Thus, the equation in standard form is:

\[
(x + 1)^2 + (y + 2)^2 = 25
\]

### Part B: Center and Radius of the Circle

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

From our standard form:

\[
(x + 1)^2 + (y + 2)^2 = 25
\]

- **Center \((h, k)\)**: The center \((h, k)\) is \((-1, -2)\).
- **Radius \(r\)**: Since \(r^2 = 25\), this implies \(r = \sqrt{25} = 5\).

**Final answer:**

- Center: \((-1, -2)\)
- Radius: \(5\)