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To transform a quadratic equation into the equation of a circle, it's crucial to use the method of completing the square. This technique is useful when equations contain squared terms, like \( x^2 \) or \( y^2 \), making them suitable for easy rewriting in a familiar circle form.

Here's a simple process to complete the square for an expression like \( x^2 - \frac{3}{2}x \):

• Take half of the coefficient of \( x \) (here it is \(-\frac{3}{2}\)), which results in \(-\frac{3}{4}\).
• Square it to get \(\left(-\frac{3}{4}\right)^2 = \frac{9}{16}\).
• Add and subtract \(\frac{9}{16}\) within the equation to balance it.

Now, the expression \( x^2 - \frac{3}{2}x \) becomes \((x - \frac{3}{4})^2 - \frac{9}{16}\). Completing the square is a vital algebraic skill, especially for simplifying and solving circle equations. Be sure to manage your equation by adding and subtracting the same number, so you maintain its equivalence.

When an equation is in the form \((x-h)^2 + (y-k)^2 = r^2\), it represents a circle with its center at the point \((h, k)\). Identifying the center of a circle allows for an understanding of its location on the coordinate plane, an essential aspect when graphing or analyzing a circle geometrically.

In the given equation, after completing the square, \[(x - \frac{3}{4})^2 + y^2 = \left(\frac{3}{4}\right)^2 \]the center can be directly identified by comparing with the standard equation of a circle. Here:

• The term \((x - \frac{3}{4})^2\) indicates that the center in the x-direction is \(\frac{3}{4}\).
• The \(y^2\) term shows that \(y-k=0\), thus \(k = 0\).

Hence, this circle's center is at \(\left(\frac{3}{4}, 0\right)\). Grasping how to find the center helps in drawing circles accurately and comprehending their position in a coordinate context.

The radius of a circle is the distance from its center to any point on the circle itself, represented by \( r \) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\). Understanding the radius provides clarity on the size of the circle.

Once you've completed the square and rearranged the equation into the standard form \[(x - \frac{3}{4})^2 + y^2 = \left(\frac{3}{4}\right)^2 \],extracting the radius is straightforward. You observe:

• The right-hand side of the equation is \(\left(\frac{3}{4}\right)^2\), indicating that \(r^2\).
• To find \(r\), simply take the square root of \(\frac{9}{16}\), which results in \(\frac{3}{4}\).

Thus, the radius of the circle is \(\frac{3}{4}\). Recognizing the radius from the equation allows you to determine the circle's size and measure proportions when mapping or visualizing circles in practical settings.