Hard

Finding the Center and Radius by Completing the Square

Simple Explanation

A sphere's equation is not always given in standard form — it may be expanded out. Completing the square on the x, y, and z terms separately converts it back into standard form, revealing the center and radius directly.

Why Do We Need It?

Real equations (from calculations or measurements) rarely come pre-packaged in standard form — this technique recovers the geometrically meaningful center and radius from whatever form the equation happens to be in.

Formula

The Standard Equation of a Sphere

(x−h)² + (y−k)² + (z−l)² = r²

A sphere is the set of all points at a fixed distance r (the radius) from a fixed center point (h,k,l).

(h,k,l): — the center of the sphere
r: — the radius of the sphere

When to use it: Whenever a sphere needs to be described by an equation, or its center and radius identified from one.

Worked Example

Find a sphere's center and radius by completing the square

Find the center and radius of the sphere x²+y²+z²−4x+6y−2z−2=0.

Why Does This Work?

Completing the square is simply a reversible algebraic rearrangement — adding and subtracting the same constant does not change an equation's solutions, so the resulting standard-form equation describes exactly the same sphere as the original expanded one.

Real-Life Example

Recovering a satellite's true coverage region from raw sensor data

Engineers derive an expanded (non-standard) equation for a satellite's signal coverage region from raw measurement data.

Completing the square converts this into standard form, revealing the actual center and radius of the coverage sphere directly.

Practice

Find the radius of the sphere x²+y²+z²+2x−4y+4=0, after completing the square.

Hard

Common mistake

Adding the completing-the-square constants only to the left side of the equation — the exact same constants must also be added to the RIGHT side, or the equation's meaning changes.

Quick Review

Group terms by variable, then complete the square for each one separately.
Add the same constants to both sides of the equation.
The result reveals the center (h,k,l) and radius r directly.