Discriminant Calculator ($\Delta = b^2 - 4ac$)
The discriminant is the key to quickly determining the nature of the solutions (roots) for any quadratic equation in the form $ax^2 + bx + c = 0$. Enter the coefficients $a$, $b$, and $c$ to find the discriminant and the full solution.
Quadratic Equation: $ax^2 + bx + c = 0$
Results
Discriminant Value ($\Delta$)
Nature of Roots
Step-by-Step Calculation
Full Quadratic Solution ($x$)
The Discriminant Formula
The discriminant, $\Delta$, is defined as the expression found beneath the square root sign in the quadratic formula:
The quadratic formula itself is used to find the roots (solutions) $x$ of a quadratic equation:
Since $\Delta$ is the value under the square root, its sign dictates whether the solutions involve real numbers (positive or zero discriminant) or imaginary numbers (negative discriminant).
How the Discriminant Determines the Roots
By simply calculating the value of $\Delta$, you can immediately know the number and type of solutions to the equation. Here are the three cases:
| Discriminant ($\Delta$) | Root Type | Graphically |
|---|---|---|
| $\Delta > 0$ (Positive) | Two Distinct Real Roots | Parabola crosses the x-axis twice. |
| $\Delta = 0$ (Zero) | One Repeated Real Root | Parabola touches the x-axis at exactly one point (the vertex). |
| $\Delta < 0$ (Negative) | Two Distinct Non-Real (Complex) Roots | Parabola never crosses the x-axis. |