Discriminant & Nature of Roots Calculator
Determine whether a quadratic has real, repeated, or imaginary roots based on its discriminant.
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Discriminant Calculator
(b² – 4ac)
Calculate the discriminant of any quadratic equation using the formula D = b² – 4ac. Instantly determine the nature of roots with a full step-by-step solution.
D = b² – 4ac
Used by students, teachers, and engineers worldwide. Our quadratic discriminant calculator handles any real coefficients a, b, and c.
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Free Calculations
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Root Types Detected
0s
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Discriminant Calculator
For equation: ax² + bx + c = 0
Discriminant (D)
0
Step-by-Step Solution
How to Use the Discriminant Calculator
Compute D = b² – 4ac in simple steps
1
Enter a, b, c
Type the three coefficients of your quadratic equation ax² + bx + c = 0 into the input fields.
2
Click Calculate
Hit the Calculate D button. The tool instantly applies the formula D = b² – 4ac.
3
Read the Result
Get the discriminant value, nature of roots, and a full step-by-step breakdown of the solution.
4
Interpret Roots
D > 0 → two real roots. D = 0 → one root. D < 0 → complex roots. All explained automatically.
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Guides, tutorials, and deep-dives for students and educators
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Basics
What Is the Discriminant? A Complete Guide
Learn what the discriminant means in algebra, where the formula D = b² – 4ac comes from, and how it unlocks the nature of roots in seconds.
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Tutorial
How to Calculate the Discriminant Step by Step
A worked example showing how to plug in a, b, and c to compute b² – 4ac correctly — with practice problems and answers included.
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Exam Prep
Discriminant on the SAT & ACT: Tips & Tricks
Standardised tests love discriminant questions. Here's how to spot them, solve them in under a minute, and avoid common mistakes.
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Advanced
Discriminant in Physics & Engineering
The discriminant isn't just classroom math — discover real-world applications in projectile motion, circuit analysis, and structural engineering.
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Concepts
Complex Roots Explained: When D < 0
When the discriminant is negative, the equation has no real solutions. Understand what complex roots are and how to express them.
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Comparison
Discriminant vs Quadratic Formula: Key Differences
The discriminant and the quadratic formula are closely related — but serve different purposes. Find out when to use each one.
Read MoreFrequently Asked Questions
Everything you need to know about the discriminant calculator
The discriminant is the expression D = b² – 4ac, where a, b, and c are the coefficients of a quadratic equation in the form ax² + bx + c = 0. It is part of the quadratic formula under the square root sign.
When D > 0, the quadratic equation has two distinct real roots. The parabola crosses the x-axis at two different points. The larger D is, the further apart the two roots are.
When D = 0, there is exactly one real root (a repeated or double root). The parabola touches the x-axis at exactly one point — its vertex lies on the x-axis.
When D < 0, the equation has two complex (imaginary) roots. There are no real solutions — the parabola does not cross the x-axis at all. The roots involve the imaginary unit i.
Yes! Our b² – 4ac calculator handles any real numbers for a, b, and c — including negative values, fractions, and decimals. The only restriction is that a ≠ 0, otherwise it would not be a quadratic equation.
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