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Completing the square is a valuable technique in algebra that is used to convert a quadratic equation into a form that makes it easier to visualize the solutions or to apply further operations, such as integrating or factoring. This method transforms polynomial expressions into perfect square trinomials, which involves creating a squared binomial from a trinomial.

• Identify the coefficients of the variable you are focusing on.
• Divide the linear coefficient by 2 and square the result.
• Add and subtract this squared number inside the equation so its value doesn't change.
• Rewrite the trinomial as the square of a binomial.

For example, to complete the square for the term with variable y, with the expression \(y^{2}-8y\), we calculate: \(8/2)^{2} = 16\), add 16 to both sides of the equation, and rewrite it as \(y-4)^{2}\). The same process is applied to the z term, which results in the equivalent expression \(z-3)^{2}\). Completing the square helps to determine the center and radius of a sphere by simplifying its equation into a standardized form.

The standard form of a sphere equation is the simplified version that clearly indicates the sphere's center and radius. A three-dimensional shape, a sphere in space, can be described using the following standard form:

\[ (x-a)^{2} + (y-b)^{2} + (z-c)^{2} = r^{2} \]

Once the equation of a sphere is in this standard form, identifying its geometric features becomes straightforward. The terms \(a, b, c\) represent the coordinates of the center of the sphere, and \(r\) denotes the radius. This form is achieved by the aforementioned technique of completing the square for each variable separately.

Going back to the example in the exercise, after completing the square for x, y, and z terms, the equation becomes \(x^{2} + (y-4)^{2} + (z-3)^{2} = 12\), which can be easily interpreted to find the center \(0, 4, 3\) and the radius \(2\sqrt{3}\).

The realm of three-dimensional coordinate geometry deals with the coordinates of points in a three-dimensional space. We use the Cartesian coordinate system, denoted generally by \(x, y, z\), to specify the location of points in 3D. The coordinates (a, b, c) define a unique point in space and serve as a reference to describe geometric figures such as lines, planes, and spheres.

In the context of spheres, the center point is essential in describing its position in space. And the distance from this center to any point on the sphere's surface, which is consistent throughout, is called the radius. This uniform distance in all directions is what defines the spherical shape. By understanding these core principles of three-dimensional geometry, students can interpret and solve problems involving 3D shapes much more effectively.