91Ó°ÊÓ

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial form. This technique is often employed to rewrite the equation of a circle so that it's easier to identify the circle's center and radius. Start by focusing on the terms involving the variables. In our exercise, we have the equation:

\( x^2 + y^2 + 4x + 2y - 20 = 0\).
First, we rearrange it to group the x and y terms together:
\( x^2 + 4x + y^2 + 2y = 20\).
Complete the square for the x and y terms:
For the x terms: \( x^2 + 4x = (x + 2)^2 - 4\).
For the y terms: \( y^2 + 2y = (y + 1)^2 - 1\).
Finally, substitute these completed squares back into the equation, and simplify to find the standard form of a circle. This step ensures the transformation to the easy-to-understand form \( (x - h)^2 + (y - k)^2 = r^2\), which helps in determining the circle's center and radius.

To locate the intercepts of the circle, you need to determine the points at which the circle crosses the x-axis and y-axis. These points are where either x or y equals zero.
Finding the x-intercepts:

• Set \( y = 0\) in the circle's equation.
• Solve the resulting equation for \( x\). In our example: \( (x+2)^2 + 1 = 25\), which simplifies to \( (x+2)^2 = 24\). Taking the square root gives \( x+2 = ±\begin { sqrt } { 24 } \), so we get the x-intercepts \( x= -2 + 2 \)

Finding the y-intercepts:

• Set \( x = 0\) in the circle's equation.
• Solve the resulting equation for \( y\). In our example: \( 4 + (y+1)^ = 25\), simplifies to \( (y+1)^2 = 21\). Taking the square root gives \( y+1 = ±\begin { sqrt } { 21 } \), giving the y-intercepts \( y = -1 ± \begin { sqrt } { 21 } \)

Understanding the intercepts can help in accurately graphing the circle and comprehending its positioning in the coordinate plane.

Graphing a circle starts with identifying its center and radius. For the equation:
\( (x+2)^2 + (y+1)^2 = 25\),
we found that:

• The center is at \( (-2, -1)\), and
• The radius is \( 5\) units.

To graph the circle:
1. Begin by plotting the center at \( (-2, -1)\) on the coordinate plane.
2. Use a compass or a round object to help draw a circle with a radius of \( 5\) units around the center.
3. Alternatively, plot points at \( 5\) units from the center in all directions (up, down, left, right). Then, sketch the curve connecting these points smoothly to form the circle.
Graphing circles correctly helps visualize their placement and how they interact with the axes, making other steps like finding intercepts more intuitive.

The standard form of a circle's equation makes it straightforward to identify key attributes, such as the center and radius. The general form is:
\( (x - h)^2 + (y - k)^2 = r^2\),
where:

• \( (h, k)\) is the center of the circle.
• \(r\) is the radius.

By converting the given equation to this form, we simplify our understanding of the circle's geometry. For example:
Our task had the equation: \( x^2 + y^2 + 4x + 2y - 20 = 0\), which we converted using Completing the Square to: \( (x + 2)^2 + (y + 1)^2 = 25\)
From this standardized form:

• The center is clearly \( (-2, -1)\).
• The radius is the square root of \( 25\), which is \( 5\).

Being able to rewrite and recognize this form is crucial for solving many circle-related problems in geometry and algebra.