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:
- Take the coefficient of \(x\) (which is \(-4\)), divide it by 2 to get \(-2\), and then square it: \((-2)^2 = 4\).
- 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:
- Take the coefficient of \(y\) (which is \(8\)), divide it by 2 to get \(4\), and then square it: \(4^2 = 16\).
- 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:
\[ (x - 2)^2 + (y + 4)^2 = 16 \]
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