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Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique can simplify solving circles stored in a standard form equation.
To complete the square, follow these steps:

• Identify the quadratic term (e.g., \( x^2 \) or \( y^2 \)) and its corresponding linear term (e.g., \( x \) or \( y \)).
• Divide the coefficient of the linear term by 2, then square the result. This value completes the square.
• Add and subtract this value inside the equation to balance it properly.

For instance, take the term \( x^2 - x \). Dividing the coefficient of \( x \) (which is -1) by 2 gives \( \frac{-1}{2} = -0.5 \). Squaring -0.5 gives \( \left(-0.5\right)^2 = 0.25 \). Add and subtract 0.25 inside the equation to complete the square: \( x^2 - x + 0.25 - 0.25 \). Rearranging, it becomes \( (x - 0.5)^2 - 0.25 \).
Similarly, applying the same steps to the y terms \( y^2 + y \) completes the square for \( y + 0.5 \), resulting in \( (y + 0.5)^2 - 0.25 \).

Finding the center of a circle from its equation involves careful algebraic manipulation. Once we have the equation in the standard form of \( (x - h)^2 + (y - k)^2 = r^2 \), the values of \( h \) and \( k \) give the coordinates of the circle's center.
Consider the initial equation: \( x^2 + y^2 - x + y + \frac{7}{2} = 0 \). Rearrange and complete the square to transform into the circle form.
Post completing the square, we obtain: \( (x - 0.5)^2 + (y + 0.5)^2 = \frac{9}{4} \)
From here, it's easy to see the center of the circle. It's given by the values \( h \) and \( k \) in the equation's perfect square form:

• \( x \text{-term}: (x - h) = (x - 0.5) \Rightarrow h = 0.5 \)
• \( y \text{-term}: (y - k) = (y + 0.5) \Rightarrow k = -0.5 \)

Therefore, the circle's center is located at \( \left( \frac{1}{2}, -\frac{1}{2} \right) \).

The radius of a circle is found once the circle's equation is in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).
Let's examine our completed square equation: \( (x - 0.5)^2 + (y + 0.5)^2 = \frac{9}{4} \).
To determine the radius \( r \), notice the constant on the right side of the equation. Here, \( r^2 = \frac{9}{4} \).
To find \( r \):

• Take the square root of both sides of the equation.
• \( r = \sqrt{\frac{9}{4}} \)
• Simplify the square root: \( r = \frac{3}{2} \)

Hence, the circle's radius is \( \frac{3}{2} \).
To visualize, remember that a radius is a line from the center of the circle (\( \left( \frac{1}{2}, -0.5 \right) \)) to its edge, with a consistent length of \( \frac{3}{2} \).