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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)

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. Warm-up: First identify exactly what the question is asking: Factor x^2 + 7x + 12.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 3 and 4 multiply to 12 and add to 7.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Core Practice: First identify exactly what the question is asking: Factor x^2 - 5x - 24.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. -8 and 3 multiply to -24 and add to -5.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Challenge: First identify exactly what the question is asking: Factor x^2 - 49.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 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: (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. Core Practice: First identify exactly what the question is asking: Factor x^2 + 11x + 30.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 5 and 6 multiply to 30 and add to 11.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Special Cases: First identify exactly what the question is asking: Factor x^2 - 10x + 25.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. It is a perfect-square trinomial.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 5)^2

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