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Completing the square is a fundamental algebraic technique used to manipulate quadratic expressions into a perfect square trinomial. This method transforms a quadratic expression, making it easier to analyze or graph. Let's see how it works with an example from the equation of a sphere.
For instance, consider the terms relating to the variable \( x \) in the given equation:

• Start with: \( x^{2} + 2x \).
• To transform this into a perfect square trinomial, take half of the coefficient of \( x \) (which is 2), then square it. This results in 1.
• Add and subtract this squared value inside the expression: \( x^{2} + 2x = (x + 1)^{2} - 1 \).

By repeating this process with each variable component in the equation, you can remodel the quadratic part of the equation into a series of perfect squares. This makes it much simpler to identify features like the center and radius of a sphere, which is essential in various mathematical and practical applications such as geometry and physics.

The center of a sphere is one of its defining characteristics in its geometric equation. In mathematical terms, this is known as the coordinate \( (h, k, l) \) in the equation of a sphere:
\[(x - h)^{2} + (y - k)^{2} + (z - l)^{2} = r^{2}\]
Once the equation is in this standard form, identifying the center is straightforward. In our completed equation:
\[(x + 1)^{2} + (y - 3)^{2} + (z - 5)^{2} = 1\]
you can observe:

• The expression \((x + 1)^{2}\) relates to \( (x - (-1))^{2} \), which indicates the x-coordinate of the center is -1.
• The expression \((y - 3)^{2}\) gives the y-coordinate, 3.
• Finally, \((z - 5)^{2}\) reveals the z-coordinate as 5.

Hence, the center of the sphere is at the point \((-1, 3, 5)\). Finding the center is crucial for understanding the sphere's position in a three-dimensional space, which is especially useful in fields like computer graphics, engineering, and beyond.

The radius of a sphere is the distance from its center to any point on its surface. It represents one of the key dimensions of a sphere.
In the standard equation of a sphere:
\[(x - h)^{2} + (y - k)^{2} + (z - l)^{2} = r^{2}\]

• The term \( r^{2} \) on the right-hand side is the square of the radius.
• From the equation of our sphere \((x + 1)^{2} + (y - 3)^{2} + (z - 5)^{2} = 1\), we can clearly see \( r^{2} = 1 \).
• Solving for \( r \), take the square root of both sides, yielding \( r = \sqrt{1} = 1 \).

Understanding the radius is critical for solving problems related to volume and surface area, among others. In practical applications like navigation and satellite technology, knowing a sphere's radius can be pivotal in calculating distances and designing pathways.