Lesson 10.3.2

Completing the Square to Graph Circles

When a circle's equation is given in general form (), you must complete the square to convert it to standard form so you can read the center and radius.

Introduction

Many problems give circle equations in expanded (general) form. You can't read the center or radius directly — you need to rewrite it by completing the square for both and .

Past Knowledge

Standard form (10.3.1). Completing the square (Algebra). Factoring perfect square trinomials.

Today's Goal

Convert general form to standard form using completing the square.

Future Success

Conic sections, ellipses, SAT/ACT problems.

Key Concepts

Completing the Square — Core Idea

Take the coefficient of , halve it, square it, and add it to both sides.

General → Standard Form


→ Complete the square for both and →

Step-by-Step Method

  1. 1Group -terms and -terms. Move the constant to the right side.
  2. 2Complete the square for the -group: add to BOTH sides.
  3. 3Complete the square for the -group: add to BOTH sides.
  4. 4Factor each perfect square trinomial. Read center and radius.

Worked Examples

Basic

Simple Case

. Find center and radius.

Group:

Complete x: . Add 9 to both sides.

Complete y: . Add 4 to both sides.

Center = (3, −2), Radius = 5

Intermediate

With Odd Coefficients

Group:

Complete x: add . Complete y: add .

Center = (−5, 1), Radius = 3

Advanced

With a Leading Coefficient

Step 0: Divide everything by 2:

Group:

Complete x: add 4. Complete y: add 9.

Center = (2, −3), Radius =

Common Pitfalls

Adding to Only One Side

When completing the square, you MUST add the same value to both sides of the equation. Forgetting to add to the right side is the #1 error.

Not Dividing by the Leading Coefficient First

If the equation has , you must divide the entire equation by 2 first. The coefficients of and must both be 1.

Real-Life Applications

Earthquake Epicenters

Seismologists write circle equations from seismic data (often in general form). Converting to standard form reveals the epicenter coordinates and the detection radius.

Satellite Orbits

Orbital equations for circular orbits start in general form. Completing the square reveals the center of the orbit (usually offset from the planet's center) and the orbital radius.

Practice Quiz

Loading...