Lesson 8.10

Difference of Squares

The "Magic Trick" of factoring. Two terms, a subtraction sign, and perfect squares. The middle term disappears.

Introduction

We've factored trinomials (3 terms). But what if there are only 2 terms, like ? This is a special case called the Difference of Squares. It's the easiest and fastest factoring method, but also the easiest to mess up if you aren't looking for it.

Past Knowledge

Lesson 8.6 (FOIL). Remember ? The middle terms cancelled out.

Today's Goal

Recognize and factor pattern.

Future Success

This pattern appears everywhere in calculus, especially in "Conjugates" for limits.

Key Concepts

The Formula

Conditions Checklist:

  • Are there exactly 2 terms?
  • Is there a Minus sign? (Difference)
  • Are both terms Perfect Squares? ()

Worked Examples

Example 1: Basic

Basic

Factor:

1

Square Root It

Square root of is .

Square root of is .

2

Write the Pattern

One plus, one minus.

Answer

Example 2: Coefficients

Intermediate

Factor:

1

Identify Squares

is the square of .

is the square of .

Answer

Example 3: GCF Hidden

Advanced

Factor:

1

Check Squared

50 is not a square. 8 is not a square.

GCF First! Pull out a 2.

2

Now Factor

becomes .

becomes .

Answer

Common Pitfalls

Sum of Squares

is PRIME. You cannot factor it. Difference means subtraction only!

Variable Powers

is NOT a difference of squares because of the power 3. Powers must be even numbers.

Real-Life Applications

Engineering Tolerances:

  • Suppose you have a square metal plate by , and you punch out a square hole by .
  • The area remaining is .
  • Knowing this factors to helps engineers calculate material stress loads more easily.

Practice Quiz

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