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Quadratics and Factoring

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

A quadratic expression has a squared variable term, usually x^2, and its graph is a parabola. Quadratics model situations with curved growth, area, projectile motion, and products of changing quantities. In Algebra I, the main goal is to recognize the structure of a quadratic, factor simple trinomials, and understand that zeros are input values where the expression equals zero. When practicing, look at the first term, middle term, and constant term. A common mistake is treating a quadratic like a linear equation and trying to solve it with only one inverse operation.

What you'll learn

Why it matters: Quadratics describe curved patterns in area, revenue, and motion when the rate itself changes instead of staying constant.

Worked example

Problem. Factor x^2 + 7x + 10.

  1. Find two numbers that multiply to 10.
  2. 5 and 2 multiply to 10 and add to 7.
  3. Write the factors as (x + 5)(x + 2).
  4. Connect the result back to Quadratics and Factoring so the method and meaning are both clear.

Answer: (x + 5)(x + 2)

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 expression is quadratic?

Choices: x^2 + 3x + 2 · 4x + 9 · 7 · x^3 + 1

Show solution
  1. A quadratic has highest power 2.
  2. x^2 + 3x + 2 is quadratic.

Answer: x^2 + 3x + 2

2. Factor x^2 + 5x + 6.

Choices: (x + 2)(x + 3) · (x + 1)(x + 6) · (x - 2)(x - 3) · (x + 5)(x + 1)

Show solution
  1. 2 and 3 multiply to 6.
  2. 2 and 3 add to 5.

Answer: (x + 2)(x + 3)

3. For x^2 + 9x + 20, what two positive numbers multiply to 20 and add to 9? Enter the larger one.

Show solution
  1. 4 and 5 multiply to 20.
  2. 4 and 5 add to 9.
  3. The larger number is 5.

Answer: 5

4. Factor x^2 + 8x + 15.

Choices: (x + 3)(x + 5) · (x + 1)(x + 15) · (x - 3)(x - 5) · (x + 2)(x + 6)

Show solution
  1. 3 and 5 multiply to 15 and add to 8.

Answer: (x + 3)(x + 5)

5. Factor x^2 - 7x + 12.

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

Show solution
  1. 3 and 4 multiply to 12 and add to 7.
  2. Both signs are negative.

Answer: (x - 3)(x - 4)

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