Discriminant Calculator (Δ = b² - 4ac)
Calculate the discriminant (b² - 4ac) of a quadratic equation and understand whether the roots are real or complex. The calculator returns a step-by-step walkthrough with exact solutions.
Quadratic Coefficients
Enter coefficients in the standard form $ax^2 + bx + c = 0$. Use decimals or common fractions (e.g., 3/2).
How to Use This Calculator
Write your quadratic equation in the form ax² + bx + c = 0. Input the numerical values for coefficients a, b, and c, then hit Calculate. The discriminant determines whether the solutions are real or complex before computing the exact values step by step.
Methodology
The calculator evaluates the expression Δ = b² - 4ac. A positive Δ produces two distinct real roots; zero results in one repeated real solution, and a negative value creates two complex conjugates. Solutions are rendered using the quadratic formula, and every step reuses consistently rounded numbers to keep the flow deterministic.
- Coefficients accept fractions (e.g.,
3/2) or decimals for flexible entry. - Step-by-step output restates the formula substitution with the exact numbers you provided.
- The exact solutions use the quadratic formula and the discriminant to keep the math transparent.
What the Discriminant Tells You
Use the sign of Δ to skip computing the square root until you know whether the roots are real or complex.
| Δ Value | Root Type | Shape |
|---|---|---|
| Δ > 0 (Positive) | Two distinct real roots | Parabola crosses the x-axis twice. |
| Δ = 0 (Zero) | One repeated real root | Parabola touches the x-axis at the vertex. |
| Δ < 0 (Negative) | Two distinct non-real (complex) roots | Parabola never crosses the x-axis. |
What is the discriminant?
The discriminant, denoted by the Greek letter Δ, is the expression b² - 4ac inside the quadratic formula. It reveals the number and nature of the roots before solving for them.
What are the three possible outcomes?
1. Δ > 0: Two real and distinct roots (parabola crosses the x-axis twice).
2. Δ = 0: One repeated real root (parabola touches the x-axis at the vertex).
3. Δ < 0: Two complex roots (parabola never crosses the x-axis).
Why is the discriminant important?
It lets you classify the solutions quickly without computing square roots. You can tell if the quadratic has real or imaginary roots just by looking at Δ.
What is a complex root?
A complex root involves the imaginary unit i = √−1. When Δ is negative, the quadratic produces two conjugate complex roots of the form a ± bi.