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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

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

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

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  1. b^2 - 4ac = 16 - 12.

Answer: 4

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

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  1. 36 - 36 = 0.

Answer: 0

3. A positive discriminant means...

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

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  1. Positive discriminants produce two real roots.

Answer: Two real solutions

4. A negative discriminant means...

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

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  1. The square root of a negative is not real.

Answer: No real solutions

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

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

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  1. It factors as (x - 2)(x - 3).

Answer: 2 and 3

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