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When solving the equation of a sphere, "completing the square" is a crucial technique. It allows us to manipulate quadratic equations into a simpler, more manageable form. Here’s how it works:
Start with a quadratic expression like \(x^2 + bx\). The goal is to turn it into a perfect square trinomial, which takes the form \((x + p)^2\). To do this, you need to find a specific number to add and subtract inside the expression.

• The first step is to divide the coefficient of \(x\) by 2. For example, if you have \(\frac{1}{2}x\) as in the exercise, divide \(\frac{1}{2}\) by 2 to get \(\frac{1}{4}\).
• Then, square this result: \((\frac{1}{4})^2 = \frac{1}{16}\).
• This squared value is then added and subtracted within the expression: \((x + \frac{1}{4})^2 - \frac{1}{16}\).

By performing this operation for each variable \(x, y, z\), the original equation becomes a sum of perfect squares that better exposes the structure of the sphere. The completion of the square simplifies the polynomial, making it easier to identify the sphere's properties such as the center and radius.

Once you have the sphere equation in its simplified form, identifying the center is straightforward. Remember, a sphere’s standard equation is \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]where \((h, k, l)\) is the center of the sphere.

• In the exercise, after completing the square, the equation becomes\[(x + \frac{1}{4})^2 + (y + \frac{1}{4})^2 + (z + \frac{1}{4})^2 = (\frac{9}{4})^2 \]
• By comparing this with the standard formula, it’s clear that \( x + \frac{1}{4} \) can be rewritten as \( (x - (-\frac{1}{4})) \). A similar transformation applies to \(y\) and \(z\).

Thus, the center \((h, k, l)\) is \(-\frac{1}{4}, -\frac{1}{4}, -\frac{1}{4}\), indicating that the sphere's center is at these coordinates.

To find the radius of a sphere, examine the equation’s form after completing the square and transform it to the standard form. Here, the sphere's equation looks like this:\[(x + \frac{1}{4})^2 + (y + \frac{1}{4})^2 + (z + \frac{1}{4})^2 = (\frac{9}{4})^2 \]

• The right side \((\frac{9}{4})^2\) indicates the square of the radius, hence the radius \(r\) is whatever value, squared, equals \((\frac{9}{4})^2\).
• Therefore, by taking the square root, we find \(r = \frac{9}{4}\).
• The radius provides the distance from the center of the sphere to any point on its surface.

Thus, the sphere has a radius\(\frac{9}{4}\), which helps visualize how large the sphere is in 3D space.