Lesson 1.18

Solving by Completing the Square

Factoring doesn't always work. Inverse operations don't work if there's an term. Completing the square is the "brute force" method to solve any quadratic equation.

Introduction

Our goal is to force the equation to look like . Once it's in that form, we can just take the square root of both sides.

Past Knowledge

You know how to complete the pattern ().

Today's Goal

Apply this to equations. Remember: What you add to one side, you MUST add to the other.

Future Success

This technique is how we prove the Quadratic Formula exists.

Key Concepts

1. The Algorithm

  1. Isolate: Move constant to the right side.
  2. Find Magic Number: .
  3. Balance: Add it to BOTH sides.
  4. Factor: Left side becomes .
  5. Solve: Square root both sides ().

2. When

Completing the square is messy if isn't 1.

You must divide EVERY term by first.

For example: becomes .

Worked Examples

Example 1: Rational Solution

Basic

Solve .

1

Move Constant & Find Magic Number

Add 7 to right. Magic number is .

2

Factor & Solve

Solutions: and

Example 2: Irrational Solution

Intermediate

Solve .

1

Setup

Move 1. Magic # is .

2

Solve

Result:

Example 3: Complex Solution

Advanced

Solve .

1

Complete Square

Move 5. Magic # is .

2

Square Root of Negative

Result:

Common Pitfalls

Unbalanced Scales

The most common error is adding the magic number to the left side but forgetting to add it to the right.

Factoring Wrong

uses . Do not use the squared value inside the parenthesis!

Real-Life Applications

Completing the square is essential for graphing circles, ellipses, and hyperbolas in Conic Sections (Chapter 10). For example, converting into center-radius form requires doing this twice (once for x, once for y).

Practice Quiz

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