Lesson 3.11

Extraneous Solutions

Sometimes the algebra produces a "solution" that makes an original denominator zero. Such an answer is called extraneous — it must be rejected.

Introduction

When we multiply by the LCD to clear denominators, we sometimes "create" solutions that didn't exist in the original equation. These are called extraneous solutions — they satisfy the cleared equation but violate the original domain restrictions.

Past Knowledge

Solving via LCD (Lesson 3.10) and domain restrictions (Lesson 3.1).

Today's Goal

Identify and reject solutions that make any original denominator zero.

Future Success

Extraneous solutions will reappear in radical equations (Unit 4) — always check!

Key Concepts

Why Do They Appear?

Multiplying both sides by the LCD is only valid when the LCD ≠ 0. If your solution makes the LCD zero, you've multiplied by zero — an illegal operation — so the result is invalid.

💡 Think of it like dividing by zero in reverse — the step that cleared the denominators was secretly dividing by zero for that particular -value.

The Check Protocol

After every rational equation:

1

List all domain restrictions FIRST

2

Solve the equation

3

Compare each solution to your restrictions

4

Reject any solution that equals a restriction

Worked Examples

Example 1: One Valid, One Extraneous

Key Example

Solve .

⚠ Domain restriction:

1

LCD = . Multiply every term.

2

Distribute and solve

🚫 is EXTRANEOUS (makes the denominator zero) → No solution

Example 2: Two Solutions, One Rejected

Intermediate

Solve .

⚠ Domain: (since )

1

Same denominator — numerators must be equal

2

Check against domain

→ makes denominator 0 → 🚫 REJECT

→ makes denominator 0 → 🚫 REJECT

🚫 Both solutions are extraneous → No solution

Example 3: Mixed — Keep One, Reject One

Advanced

Solve .

⚠ Domain:

1

LCD = . Multiply every term.

2

Solve

3

Check: is not 0 or −2

✓

Common Pitfalls

Assuming All Solutions Are Valid

The #1 mistake in rational equations! Always check every solution against the original domain restrictions.

Writing "No Solution" Too Quickly

If you get two solutions and one is extraneous, the other might still be valid. Only write "no solution" when ALL solutions fail the check.

Real-Life Applications

In engineering, mathematical models may produce solutions that are physically impossible (like a negative length or infinite speed). The discipline of checking for extraneous solutions trains you to always validate mathematical results against real-world constraints.

Practice Quiz

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