91Ó°ÊÓ

An equation that represents a circle is fundamentally a way to describe all the points that maintain a constant distance, known as the radius, from a central point. The standard equation of a circle with a center at

• ext{h}
• ext{k}

is given by:\[(x-h)^2 + (y-k)^2 = r^2\] where

• ext{r} is the radius of the circle.
• ext{h}, ext{k} are the coordinates of the center of the circle.

In our exercise, by completing the square, we modify the original equation into this standard form. The benefits of having it in this form are immense as it directly shows us the circle's center and its radius, making it easier to plot or visualize.
Rewriting any circle's equation into this form clarifies its geometric features, simplifying computations and understanding.

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It's crucial when working with quadratic equations and transformations.
For the expression to be a perfect square trinomial, it should follow the form \[(a imes x + b)^2\] which expands to \[a^2 imes x^2 + 2ab imes x + b^2.\]
In the problem, we face:\[x^2 - 4x. \]
This portion can be rearranged by completing the square. The steps include:

• Identifying the coefficient of x, which is -4 in this case.
• Taking half of this value: -4/2 = -2.
• Squaring the result: (-2)^2 = 4.

So, we rewrite \[x^2 - 4x + 4\]to create a perfect square trinomial: \[(x-2)^2.\]This rewriting into a perfect square trinomial helps to transform quadratic equations into forms that are easier to understand and work with, especially when plotting curves like circles.

Determining the center and radius of a circle from its equation allows one to quickly understand its geometric properties. Once the equation is in the form \[(x-h)^2 + (y-k)^2 = r^2,\]the process is straightforward.

• The center of the circle is given by the coordinates \[(h, k).\]
• The radius is equal to the square root of \[r^2.\]

In our adjusted equation, we have: \[(x-2)^2 + y^2 = 4.\]
This translates to a circle centered at the point \[(2, 0)\] with a radius of \[ ext{2}.\]Having this descriptive detail makes it easy to plot the circle on a graph or explore its properties, such as intercepts or tangents. The visualization of a circle through its center and radius aids in both comprehension and computational applications.