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Completing the square is an essential method used to transform a quadratic equation into a perfect square trinomial. This technique is particularly helpful when working with equations of a circle, as it allows us to rewrite them in a form that easily reveals their geometric properties. Let's take a look at how to apply this method.

• First, focus on the quadratic terms, such as the terms involving only x or y.
• Extract the coefficient of the linear term (the term with just x or y), divide it by two, and then square the result.
• Introduce this squared term to both sides of the equation, ensuring that the balance is maintained.

For example, consider the equation from the problem: \[x^2 - \frac{1}{2}x + \frac{1}{16} - \frac{1}{16}\]It can be transformed into:\((x - \frac{1}{4})^2\).This completed square form allows us to see the grouped variable and its relationship to determine the circle's properties.

The center and radius of a circle provide vital information about its position and size in coordinate geometry. The equation of a circle in standard form is expressed as:\((x - h)^2 + (y - k)^2 = r^2\)where:

• h and k represent the coordinates of the center
• r is the radius of the circle

By comparing this with the completed square form, the values for the center and radius can be easily identified without complicated calculations. From our example, \((x - \frac{1}{4})^2 + (y + \frac{1}{4})^2 = \frac{3}{8}\),You can see the center is at \(\left( \frac{1}{4}, -\frac{1}{4} \right)\),while the radius is determined as \(\sqrt{\frac{3}{8}}\).These contextual interpretations help visualize the circle's location and coverage.

The equation of a circle is typically written in standard form as \((x - h)^2 + (y - k)^2 = r^2\),representing all points \((x, y)\)that are a distance r from the center \((h, k)\).

Steps to identifying and using this form:

• Whenever given an equation that looks quadratic in both x and y, consider if it represents a circle.
• Rearrange and complete the square for both variables to easily transition it into the standard form.
• Compare your equation with the standard circle equation format to extract specific details like the center and the radius.

For the exercise, showing that the provided equation represents a circle involved grouping terms and completing the square for both variables, x and y, aligning them neatly into the circle’s standard form. This fundamental skill not only identifies circular equations but also enriches understanding of graphically placing such shapes within coordinate systems.