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Quadratics: Factoring

A free College Algebra lesson from the “Quadratics: Multiplying and Factoring” unit, with a worked example and practice problems including step-by-step solutions.

Factoring a quadratic rewrites it as a product of simpler expressions. For trinomials like x^2 + bx + c, the goal is to find two numbers that multiply to c and add to b. Factoring matters because it reveals zeros, supports graphing, and prepares for solving equations with the zero-product property. When practicing, look for a greatest common factor first, then identify the trinomial pattern. A common mistake is stopping when the constant factors are correct without checking the middle term. Always expand your factors to verify the original expression.

What you'll learn

Why it matters: Break-even points, area products, and zero-finding problems often become easier after a quadratic is factored. Factoring turns a dense expression into two simpler pieces whose product structure can be checked by expanding.

Worked example

Problem. Factor x^2 + 9x + 20.

  1. Find factors of 20.
  2. 4 and 5 multiply to 20 and add to 9.
  3. Write (x + 4)(x + 5).

Answer: (x + 4)(x + 5)

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. Factor x^2 + 7x + 12.

Choices: (x + 3)(x + 4) · (x + 1)(x + 12) · (x - 3)(x - 4) · (x + 6)(x + 2)

Show solution
  1. 3 and 4 multiply to 12 and add to 7.

Answer: (x + 3)(x + 4)

2. Factor x^2 - 5x - 24.

Choices: (x - 8)(x + 3) · (x + 8)(x - 3) · (x - 6)(x + 4) · (x - 24)(x + 1)

Show solution
  1. -8 and 3 multiply to -24 and add to -5.

Answer: (x - 8)(x + 3)

3. Factor x^2 - 49.

Choices: (x - 7)(x + 7) · (x - 49)(x + 1) · (x + 7)^2 · (x - 7)^2

Show solution
  1. Difference of squares.

Answer: (x - 7)(x + 7)

4. Factor x^2 + 11x + 30.

Choices: (x + 5)(x + 6) · (x + 10)(x + 3) · (x - 5)(x - 6) · (x + 30)(x + 1)

Show solution
  1. 5 and 6 multiply to 30 and add to 11.

Answer: (x + 5)(x + 6)

5. Factor x^2 - 10x + 25.

Choices: (x - 5)^2 · (x + 5)^2 · (x - 25)(x + 1) · (x - 10)(x + 25)

Show solution
  1. It is a perfect-square trinomial.

Answer: (x - 5)^2

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