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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.

Practice problems

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

Choices: Radical · Linear · Constant · Arithmetic sequence

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which equation type often creates extraneous solutions when squared?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Squaring can introduce extra solutions.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Radical

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

Choices: -6 · 6 · 0 · 1

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which value is excluded from 1/(x + 6)?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. x + 6 cannot equal 0.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: -6

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

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: For x^2 - 9 = 0, a good method is...
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. It is a difference of squares.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Factoring

4. After clearing denominators, you should...

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

Show solution
  1. Challenge: First identify exactly what the question is asking: After clearing denominators, you should...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Clearing denominators can hide excluded values.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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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