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Solving Quadratics by Factoring

A free Algebra I lesson from the “Quadratic Foundations” unit, with a worked example and practice problems including step-by-step solutions.

When a quadratic is written as a product equal to zero, at least one factor must be zero. Factoring changes the problem into simpler linear equations. In Quadratic Foundations, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.

What you'll learn

Why it matters: The zero product property helps find break-even points, launch and landing times, and dimensions that make an area model hit a target value.

Worked example

Problem. Solve x^2 - 5x + 6 = 0.

  1. Factor the quadratic: x^2 - 5x + 6 = (x - 2)(x - 3).
  2. Set each factor equal to zero.
  3. x = 2 or x = 3.
  4. Connect the result back to Solving Quadratics by Factoring so the method and meaning are both clear.

Answer: x = 2 or x = 3

Practice problems

1. Solve (x - 4)(x + 2) = 0.

Choices: 4 and -2 · -4 and 2 · 4 and 2 · -4 and -2

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve (x - 4)(x + 2) = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set each factor equal to zero.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 4 and -2

2. Solve (x + 5)(x + 1) = 0.

Choices: -5 and -1 · 5 and 1 · -5 and 1 · 5 and -1

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve (x + 5)(x + 1) = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x + 5 = 0 gives -5 and x + 1 = 0 gives -1.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: -5 and -1

3. Solve x^2 - 9 = 0. Enter the positive solution.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 - 9 = 0. Enter the positive solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x - 3)(x + 3).
  4. The positive solution is 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

4. Solve x^2 + 7x + 10 = 0.

Choices: -5 and -2 · 5 and 2 · -10 and -1 · 5 and -2

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x^2 + 7x + 10 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x + 5)(x + 2).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: -5 and -2

5. Solve x^2 - 7x + 12 = 0.

Choices: 3 and 4 · -3 and -4 · 2 and 6 · 1 and 12

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x^2 - 7x + 12 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x - 3)(x - 4).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
  5. Identify the Algebra I structure before choosing a calculation.

Answer: 3 and 4

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