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Equation Strategy and Extraneous Solutions

A free Algebra II lesson from the “Equations, Rational Functions, and Conics” unit, with a worked example and practice problems including step-by-step solutions.

Algebra II equations come in many forms. The first move is to name the structure, choose a method, and track restrictions so extraneous solutions do not slip in.

What you'll learn

Why it matters: Real problems rarely tell you what kind of equation they are. Choosing the right tool — factor, isolate, square, log, substitute — and checking for extraneous solutions is the most-used Algebra II skill, especially on standardized tests.

Worked example

Problem. Why must x = 2 be rejected for 1/(x - 2) = 5?

  1. The denominator is x - 2.
  2. When x = 2, the denominator is 0.
  3. Division by zero is undefined, so x = 2 is not allowed.

Answer: It makes the denominator zero.

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Which equation type often creates extraneous solutions when squared?

Choices: Radical · Linear · Constant · Arithmetic sequence

Show solution
  1. Squaring can introduce extra solutions.

Answer: Radical

2. Which value is excluded from 1/(x + 6)?

Choices: -6 · 6 · 0 · 1

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  1. x + 6 cannot equal 0.

Answer: -6

3. For x^2 - 9 = 0, a good method is...

Choices: Factoring · Logarithms · Unit circle · Synthetic division only

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  1. It is a difference of squares.

Answer: Factoring

4. After clearing denominators, you should...

Choices: Check in the original equation · Never check · Ignore restrictions · Only graph

Show solution
  1. Clearing denominators can hide excluded values.

Answer: Check in the original equation

5. Why should a solution be checked after squaring both sides?

Choices: Squaring can create extraneous solutions · Squaring always loses every solution · Checking changes the equation · Only linear equations need checks

Show solution
  1. Squaring is not always reversible.
  2. A candidate can satisfy the squared equation but not the original.
  3. That is why checking matters.

Answer: Squaring can create extraneous solutions

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