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Solving Quadratic Equations

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

Quadratic equations can be solved by factoring, square roots, completing the square, or the quadratic formula. The best method depends on the form.

What you'll learn

Why it matters: Quadratic equations appear in height, area, profit, and design constraints, but not every one should be solved the same way. Seeing the structure first helps students choose factoring, square roots, or the quadratic formula deliberately.

Worked example

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

  1. Factor as (x - 3)(x - 4) = 0.
  2. Set each factor equal to zero.
  3. x = 3 or x = 4.

Answer: x = 3 or x = 4

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. Solve x^2 - 9 = 0.

Choices: x = 3 or x = -3 · x = 9 · x = -9 · No real solution

Show solution
  1. x^2 = 9.

Answer: x = 3 or x = -3

2. Solve x^2 + x - 20 = 0.

Choices: x = 4 or x = -5 · x = -4 or x = 5 · x = 2 or x = 10 · No real solution

Show solution
  1. Factor as (x + 5)(x - 4).

Answer: x = 4 or x = -5

3. Find the discriminant of x^2 - 6x + 9 = 0.

Show solution
  1. 36 - 36 = 0.

Answer: 0

4. Solve x^2 = 49.

Choices: x = 7 or x = -7 · x = 7 only · x = -7 only · No real solution

Show solution
  1. Both 7 and -7 square to 49.

Answer: x = 7 or x = -7

5. Solve x^2 - 9x = 0.

Choices: x = 0 or x = 9 · x = -9 only · x = 9 only · x = -3 or x = 3

Show solution
  1. Factor x(x - 9).

Answer: x = 0 or x = 9

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