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Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This approach makes it easier to graph ellipses and circles as it allows us to rewrite the equation in a standardized form. Consider the equation from our exercise:
\[ x^2 - \frac{3}{2}x + y^2 = 0 \]To complete the square for the \( x \) terms, you identify the linear coefficient, which is \(-\frac{3}{2}\) in this case. Divide it by 2, resulting in \(-\frac{3}{4}\). Squaring \(-\frac{3}{4}\) yields \( \frac{9}{16} \).
You then add and subtract this squared value inside the expression:

• Add \( \frac{9}{16} \) to complete the square.
• Subtract \( \frac{9}{16} \) to maintain the balance of the equation.

This step ensures the \( x \) terms can be rewritten as a perfect square:
\[ \left( x - \frac{3}{4} \right)^2 \]
Completing the square transforms parts of the equation into a form that reveals geometrical properties more clearly. Next, we can find the center of the circle.

The center of a circle in its standard equation form \((x - h)^2 + (y - k)^2 = r^2\)is denoted by the point \((h, k)\). In geometrical terms, this point is equidistant from all points along the circle's boundary.
From the equation \((x - \frac{3}{4})^2 + y^2 = \frac{9}{16}\), we can identify \(h = \frac{3}{4}\) and \(k = 0\). This tells us that:

• The center is located at the point \(\left(\frac{3}{4}, 0\right)\).
• This position means the circle is shifted horizontally by \(\frac{3}{4}\) units from the origin along the x-axis.

Finding the center is critical as it helps in understanding the exact position of your circle in the coordinate plane, ensuring you can graph it accurately.

The radius of a circle is defined as the distance from the center of the circle to any point on its circumference. This measurement is a constant for any circle and is denoted by \( r \).
In the standard circle formula \((x - h)^2 + (y - k)^2 = r^2\), \(r^2\) is explicitly given. For our example:

• \(r^2 = \frac{9}{16}\)
• To find the radius \(r\), take the square root of both sides, resulting in \(r = \frac{3}{4}\).

Knowing the radius allows you to understand the scale and size of the circle in comparison to other elements in a graph. Its determination is crucial for accurate depiction and analysis of circular shapes in various mathematical problems.