Two Distinct Real Roots
How to Use the Calculator
Get your diagnostic report in two simple steps.
Input Coefficients
Enter the values for a, b, and c into the equation composer at the top. You can also use the sliders for interactive exploration.
Analyze the Results
The diagnostic report and the parabola plot update in real-time, showing the discriminant value (Δ) and the nature of the equation's roots.
What is the Discriminant?
The key to unlocking the nature of a quadratic equation's roots.
In algebra, the discriminant of a quadratic equation ax² + bx + c = 0 is a value that helps "discriminate" or determine the number and type of solutions (or roots) the equation will have, without having to solve the equation completely.
It is the expression found under the square root sign in the quadratic formula. The formula for the discriminant is:
The value of delta (Δ) tells you everything you need to know about the roots, which our quadratic equation discriminant calculator visualizes for you.
Interpreting the Diagnostic Report
Understanding what the value of the discriminant means.
Δ > 0 (Positive)
The equation has two distinct real roots. This means the parabola on the graph will intersect the x-axis at two different points.
Δ = 0 (Zero)
The equation has one repeated real root. This means the vertex of the parabola will touch the x-axis at exactly one point.
Δ < 0 (Negative)
The equation has two complex conjugate roots (and no real roots). This means the parabola will not intersect the x-axis at all.
Frequently Asked Questions
Common questions about the quadratic discriminant.
Why is the discriminant important?
It provides a quick "health check" of a quadratic equation. Before spending time to solve for the roots, you can use the discriminant to know what kind of answer to expect. This is critical in fields like physics and engineering, where the nature of the roots (e.g., real vs. complex) determines the behavior of a system.
Can 'a' be zero in a quadratic equation?
No. If the coefficient 'a' is zero, the ax² term vanishes, and the equation becomes a linear equation (bx + c = 0), not a quadratic one. Our calculator allows 'a' to be zero to show how the parabola becomes a straight line.
Where does the discriminant formula come from?
It comes directly from the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. The discriminant is the part inside the square root, b² - 4ac. The square root of a positive number is real, the square root of zero is zero, and the square root of a negative number is imaginary, which directly explains the nature of the roots.
