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Completing the square is a method used to convert a quadratic equation into a perfect square trinomial. This helps in rewriting equations in a more useful form, such as the standard form of a circle. To complete the square, you add and subtract a specific value so that the expression inside parentheses forms a perfect square.For example, consider the equation: $$ x^2 + 2x $$To complete the square, you take half of the coefficient of x (which is 2), square it to get 1, then add and subtract this value within the equation:$$ x^2 + 2x + 1 - 1 $$This can be rewritten as:$$ (x + 1)^2 - 1 $$Similarly, for $$ y^2 - 6y $$take half of -6 (-3), square it to get 9, and add and subtract 9:$$ y^2 - 6y + 9 - 9 $$This can be rewritten as:$$ (y - 3)^2 - 9 $$By completing the square, quadratic terms are transformed into more manageable forms, making it easier to work with equations, particularly to convert them into the standard form of a circle.

The standard form of a circle’s equation is$$ (x - h)^2 + (y - k)^2 = r^2 $$This equation represents a circle with a center at $$ (h, k) $$and a radius $$ r $$To convert a given equation into the standard form, it may be necessary to use the completing the square method.For the given exercise, after completing the square for both x and y terms, the equation transforms from:$$ x^2 + y^2 + 2x - 6y = -6 $$To:$$ (x + 1)^2 + (y - 3)^2 = 4 $$This is now in standard form. From this, you can identify:

• The center, $$ (h, k) = (-1, 3) $$
• The radius, $$ r = \sqrt{4} = 2 $$

Rewriting the equation in standard form makes it straightforward to determine these characteristics of the circle and is essential for graphing the circle accurately.

Graphing a circle becomes straightforward once you have the equation in standard form. Here’s a step-by-step guide:1. **Identify the Center and Radius**:For the equation$$ (x + 1)^2 + (y - 3)^2 = 4 $$,we see that the center is at $$ (-1, 3) $$and the radius is $$ 2 $$2. **Plot the Center**:Mark the center of the circle on the coordinate plane at $$ (-1, 3) $$3. **Draw the Circle**:Using the radius, measure 2 units from the center in all directions: up, down, left, and right. These points are the boundary points of your circle.4. **Connect the Points**:Use a smooth, round curve to connect these boundary points, effectively forming the circle.Graphing circles by following these clear steps helps visualize the mathematical relationship defined by the equation and provides a clear picture of the circle’s placement in the coordinate plane.