Part A: Complete the square to rewrite the following equation in standard form. Show all necessary work.

Question

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: \(5\)