CMClearMathAcademy

Quadratic Formula and Discriminant

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

The quadratic formula solves any quadratic equation ax^2 + bx + c = 0. The discriminant, b^2 - 4ac, is the part under the square root and tells what kind of solutions the equation has. A positive discriminant gives two real solutions, zero gives one repeated real solution, and a negative discriminant gives two complex solutions. This matters because not every quadratic factors nicely. When practicing, identify a, b, and c carefully, substitute them into the formula, and simplify in stages. A common mistake is losing the negative sign on b or forgetting parentheses around negative values.

What you'll learn

Why it matters: Projectile motion, profit maximization, and engineering tolerance work all reduce to solving a quadratic. The discriminant tells you how many real solutions exist before you compute them, so you can spot 'no real answer' or 'one repeated root' situations early.

Worked example

Problem. For x^2 - 6x + 8 = 0, find the discriminant.

  1. Here a = 1, b = -6, c = 8.
  2. Compute b^2 - 4ac.
  3. 36 - 32 = 4.

Answer: 4

Practice problems

1. Find the discriminant of x^2 - 4x + 3 = 0.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the discriminant of x^2 - 4x + 3 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. b^2 - 4ac = 16 - 12.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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

Show solution
  1. Warm-up: 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

3. A positive discriminant means...

Choices: Two real solutions · One real solution · No real solutions · No x-intercepts ever

Show solution
  1. Core Practice: First identify exactly what the question is asking: A positive discriminant means...
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. Positive discriminants produce two real roots.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Two real solutions

4. A negative discriminant means...

Choices: No real solutions · Two real solutions · One repeated real solution · A linear equation

Show solution
  1. Core Practice: First identify exactly what the question is asking: A negative discriminant means...
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The square root of a negative is not real.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No real solutions

5. Solve x^2 - 5x + 6 = 0.

Choices: 2 and 3 · -2 and -3 · 1 and 6 · No real solutions

Show solution
  1. Challenge: First identify exactly what the question is asking: Solve x^2 - 5x + 6 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. It factors as (x - 2)(x - 3).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2 and 3

Practice this interactively with instant feedback and an AI tutor.

Practice Quadratic Formula and Discriminant Take the free placement check

More Algebra II lessons