Discriminant Calculator
Find the discriminant of a quadratic and its roots.
Results update as you type.
How it works
The discriminant of a quadratic equation instantly tells you how many real solutions it has without solving it, and this calculator computes it. You enter the coefficients a, b, and c of a quadratic in the form ax squared plus bx plus c, and it returns the discriminant and a description of the roots. The discriminant is the expression b squared minus 4ac, which is the part under the square root in the quadratic formula. Its sign is what matters: when the discriminant is positive, the square root is a real number and the equation has two distinct real roots; when it is exactly zero, the square root vanishes and there is one repeated real root, meaning the parabola just touches the x-axis; and when it is negative, the square root of a negative number is not real, so there are no real roots, only a pair of complex conjugate solutions. This makes the discriminant a quick diagnostic tool in algebra, useful for understanding a parabola's graph, checking whether an equation can be solved with real numbers, and analyzing problems in physics and engineering. The coefficient a must not be zero, since with a equal to zero the equation is linear, not quadratic, and has no discriminant. Enter a, b, and c to find the discriminant and the nature of the roots.
Discriminant = b² − 4ac
For a quadratic ax² + bx + c, the discriminant is b squared minus 4ac. Its sign reveals the roots: a positive discriminant means two distinct real roots, zero means one repeated real root, and a negative value means no real roots, only a complex conjugate pair. It is the term under the square root in the quadratic formula.
Source:Algebra
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter a (coefficient of x²), b (coefficient of x) and c (constant).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Discriminant (b² − 4ac) updates instantly as you type — no button to press.
Frequently asked questions
What is the discriminant?
The discriminant is the expression b squared minus 4ac from a quadratic equation ax squared plus bx plus c. It is the quantity under the square root in the quadratic formula, and its sign reveals how many real roots the equation has without requiring you to solve the equation fully. It is a fundamental diagnostic in algebra.
What does the discriminant tell you about roots?
A positive discriminant means the quadratic has two distinct real roots. A discriminant of exactly zero means there is one repeated real root, where the parabola touches the x-axis at a single point. A negative discriminant means there are no real roots; the solutions are a pair of complex conjugate numbers instead. The sign alone determines the nature of the roots.
Why is the discriminant under the square root?
In the quadratic formula, the solutions are negative b plus or minus the square root of the discriminant, all over 2a. Because you take the square root of the discriminant, its sign controls whether that root is real. A negative value has no real square root, which is why a negative discriminant produces complex rather than real solutions.
Why must the coefficient a not be zero?
If a is zero, the x squared term disappears and the equation becomes linear, bx plus c equals zero, rather than quadratic. A linear equation does not have a discriminant because it has at most one solution and no parabola. The discriminant is defined only for genuine quadratics, so the calculator requires a to be nonzero.
How is the discriminant used in graphing?
The discriminant tells you how a parabola relates to the x-axis. A positive value means the parabola crosses the axis at two points, zero means it touches at exactly one point (the vertex sits on the axis), and a negative value means it never touches the axis. This helps you sketch or interpret a quadratic's graph quickly.