Lesson 1.9

Solving by Factoring

We turn the skill of factoring into a solving superpower. If you can break an equation into pieces that equal zero, you can find the solution instantly.

Introduction

The Zero Product Property is one of the most useful rules in algebra. It states a simple truth: if two numbers multiply to zero, one of them must be zero. This lets us solve complex quadratic equations by breaking them into simple linear ones.

Past Knowledge

You know how to factor into .

Today's Goal

We use that factorization to solve equations like by finding where each factor equals zero.

Future Success

This technique finds the exact moment a projectile hits the ground or a business breaks even.

Key Concepts

The Zero Product Property

If , then:

This ONLY works when the equation equals zero. You cannot say "if , then ". That is false!

The Strategy

  1. Set to Zero: Move everything to one side so the equation equals 0.
  2. Factor Completely: GCF, then Trinomials.
  3. Split & Solve: Set each specific factor to 0 and solve for x.

Worked Examples

Example 1: Solve by Factoring

Basic

Solve .

1

Factor the Trinomial

Find numbers that multiply to 6 and add to 5. (2 and 3).

2

Set Each to Zero

Example 2: Solve by Grouping ()

Intermediate

Solve .

1

AC Method

. Add to 7.
Factors: 8 and -1.

Split: .

Group: .

2

Solve

Solutions:

Example 3: Not Set to Zero

Advanced

Solve .

1

Set to Zero First!

Move and to the left.

2

Factor and Solve

Multiply to -12, Add to -4. (-6 and 2).

Common Pitfalls

Dividing by x

For , NEVER divide by x. You will lose the solution . Instead, subtract and factor: .

Forgetting to Set to Zero

If , you cannot say . That is wrong! You must FOIL, move the 8 over to get =0, and then re-factor.

Real-Life Applications

Projectile Motion (Landing)

When does a rocket hit the ground? The ground is height .

If , we solve . Factoring out gives . So (launch) and (landing).

Practice Quiz

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