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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

Practice problems

1. Solve x^2 - 9 = 0.

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x^2 - 9 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x^2 = 9.
  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 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. Core Practice: First identify exactly what the question is asking: Solve x^2 + x - 20 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor as (x + 5)(x - 4).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 4 or x = -5

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

Show solution
  1. Challenge: First identify exactly what the question is asking: Find the discriminant of x^2 - 6x + 9 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 36 - 36 = 0.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

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. Square Roots: First identify exactly what the question is asking: Solve x^2 = 49.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. Both 7 and -7 square to 49.
  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 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. Factoring: First identify exactly what the question is asking: Solve x^2 - 9x = 0.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Factor x(x - 9).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 0 or x = 9

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