Discriminant Calculator
Enter the coefficients a, b and c from a quadratic equation to find its discriminant and what it says about the roots.
The discriminant of a quadratic ax² + bx + c is b² − 4ac. If it is positive the equation has two distinct real roots; if it is zero there is one repeated real root; if it is negative there are two complex conjugate roots. It tells you the nature of the roots without solving the equation.
How to Use This Calculator
Write your equation in standard form, ax² + bx + c = 0, and read off the three coefficients. The coefficient a is the number in front of x², b is in front of x, and c is the constant. Enter each into the matching field.
The calculator shows the discriminant and interprets it, and the step-by-step working shows the substitution so you can reproduce it by hand. If a is zero the equation is not quadratic, and the discriminant does not apply.
How the Calculation Works
The discriminant is the part of the quadratic formula that sits under the square root sign. Because you cannot take the real square root of a negative number, its sign alone decides whether the roots are real or complex, without solving the whole equation.
A positive discriminant means the square root is a real number, giving two different real roots. Zero means the square-root term vanishes, leaving one repeated root. A negative value means the square root is imaginary, giving a complex conjugate pair.
Δ = b² − 4ac
Worked Example
Take the equation x² + 5x + 6 = 0, so a = 1, b = 5 and c = 6. Substitute into the formula: b² is 5² = 25, and 4ac is 4 × 1 × 6 = 24.
The discriminant is 25 − 24 = 1. Because it is positive, the equation has two distinct real roots. (They factor as (x + 2)(x + 3), giving roots −2 and −3.)
What the Result Means
The discriminant tells you what kind of solutions to expect before you solve. A positive value promises two real answers, which is what you want when a problem should have two distinct solutions. Zero signals a repeated root, where the parabola just touches the x-axis. A negative value means the parabola never crosses the x-axis and the solutions are complex.
A positive discriminant that is a perfect square, like 1, 4 or 9, also tells you the quadratic factors neatly over the integers.
Assumptions and Limitations
- The equation must be quadratic — a cannot be zero
- Coefficients can be negative or decimal; enter signs carefully
- The discriminant describes the nature of the roots, not their values
- A perfect-square positive discriminant indicates the quadratic factors over the integers
Who Uses a Discriminant Calculator
Algebra and precalculus students use this to tell how many real solutions a quadratic has before solving it. Check the sign of b² − 4ac to know whether a parabola crosses the x-axis twice, touches once, or not at all. It is a fast homework check and a way to understand the nature of a quadratic's roots without grinding through the full formula.
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Open calculator →Frequently Asked Questions
What does the discriminant tell you?
It tells you the nature of a quadratic's roots without solving it. Positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots.
What if the discriminant is zero?
The quadratic has exactly one real root, repeated twice. Graphically, the parabola touches the x-axis at a single point rather than crossing it.
What does a negative discriminant mean?
There are no real roots. The two solutions are complex conjugates involving the imaginary unit i, and the parabola never crosses the x-axis.
Can the discriminant tell me if a quadratic factors?
Yes. If the discriminant is a positive perfect square, such as 1, 4 or 9, the quadratic factors neatly over the integers. If it is positive but not a perfect square, the roots are real but irrational.
Why is a not allowed to be zero?
If a is zero there is no x² term, so the equation is linear, not quadratic. The discriminant b² − 4ac is only defined for quadratic equations.
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