Use this discriminant calculator to compute b² − 4ac, classify quadratic roots, see steps, and interpret real, repeated, or complex roots.
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Quadratic discriminant and root-type check Enter the standard-form coefficients from ax² + bx + c = 0. The calculator shows Δ = b² − 4ac, the substitution work, root type, roots, vertex, and x-axis interpretation.
Inputs
Coefficient entry
Use signs exactly as written. Decimals and simple fractions are accepted, so an input such as 1/2 is read as 0.5 before the discriminant formula is applied.
Quick examples
Result
Δ = 1
x² − 5x + 6 = 0
Root nature
Two distinct real roots
Factorability cue
Perfect square
Root x₁
3
Root x₂
2
Vertex
(2.5, -0.25)
Step
Value
Substitute
Δ = (-5)² − 4(1)(6)
Compute terms
b² = 25 and 4ac = 24
Finish
Δ = 25 − 24 = 1
Graph meaning
The parabola crosses the x-axis at two different points.
Interpretation
A positive discriminant gives two distinct real roots. Perfect-square discriminant; integer coefficients can give rational roots.
Discriminant calculator for quadratic root types, steps, and graph meaning
A discriminant calculator computes Δ = b² − 4ac for a quadratic equation ax² + bx + c = 0 and uses the result to classify the roots. The sign of the discriminant tells you whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots, while the step table and vertex check help you catch sign errors before you trust the answer.
Interpreting the discriminant
The discriminant Δ = b² − 4ac appears under the square root in the quadratic formula: x = (−b ± √Δ) / (2a). Its sign determines what lies under that radical.
When Δ > 0, the square root is real and positive, giving two distinct real roots. When Δ = 0, the square root vanishes, giving one repeated root x = −b/(2a). When Δ < 0, the square root of a negative number produces complex conjugate roots.
This page keeps the discriminant value beside the root classification because many algebra mistakes are sign mistakes. Seeing b², 4ac, and the final subtraction separately makes it easier to check whether a negative coefficient was squared correctly or whether subtracting a negative 4ac term should have become addition.
Δ = b² − 4ac
The discriminant of a quadratic equation.
x = (−b ± √Δ) / (2a)
Quadratic formula using the discriminant.
How to use the discriminant calculator
Start by rewriting the equation in standard form, with every term on one side and zero on the other side. Then copy the coefficients into a, b, and c. For example, 2x² − 7x + 3 = 0 has a = 2, b = −7, and c = 3.
The calculator accepts decimals and simple fractions, then substitutes those values into b² − 4ac. The result sheet shows the substitution line, the b² and 4ac terms, the final discriminant, the root type, the roots when they are meaningful for the equation type, and the graph meaning.
If a = 0, the equation is not quadratic. The calculator treats it as a linear or no-equation case rather than pretending the quadratic discriminant still applies. That warning is useful when the x² term was accidentally dropped during rearrangement.
Positive, zero, and negative discriminants
A positive discriminant means the quadratic has two distinct real roots. On the graph, the parabola crosses the x-axis twice. If the discriminant is also a perfect square and the coefficients are integers, the roots are rational, which often means the quadratic can be factored neatly over the rationals.
A zero discriminant means the two quadratic-formula branches collapse to the same value. The equation has one repeated real root, and the parabola touches the x-axis exactly at the vertex.
A negative discriminant means the square root term is not real. The roots are a complex conjugate pair, the real part sits on the axis of symmetry, and the corresponding parabola has no real x-intercepts.
Δ > 0: two distinct real roots.
Δ = 0: one repeated real root.
Δ < 0: two complex conjugate roots.
Δ is a non-negative perfect square: integer-coefficient quadratics can have rational roots.
Worked example: 2x² − 7x + 3 = 0
For 2x² − 7x + 3 = 0, the coefficients are a = 2, b = −7, and c = 3. Substitute them into the discriminant formula: Δ = (−7)² − 4(2)(3).
That gives Δ = 49 − 24 = 25. Because 25 is positive, the equation has two distinct real roots. Because 25 is a perfect square, the roots are rational: x = (7 ± 5) / 4, so x = 3 and x = 1/2.
The same result has a graph meaning. The parabola opens upward because a is positive, crosses the x-axis twice, and has an axis of symmetry halfway between the two roots at x = 7/4.
Discriminant versus full quadratic formula
The discriminant is not the whole quadratic formula. It is the expression under the radical. That makes it fast for classifying the nature of roots, checking whether factoring is likely, and deciding whether the graph crosses the x-axis.
The full quadratic formula is still needed when you want the exact root values. This calculator includes both views so the page works as a quadratic discriminant calculator and a quick root-type checker without forcing you to open a separate quadratic formula page for every problem.
If you need complete exact radical simplification, vertex form, intercept form, or a broader graph worksheet, the related quadratic formula and completing-the-square calculators are better fits. The discriminant page stays focused on the fastest classification workflow.
Common discriminant mistakes
The most common setup error is copying b without its sign. In x² − 5x + 6, b is −5, not 5. Squaring b makes the sign disappear in b², but the sign still matters in the root formula and in other graph features.
Another common mistake is mishandling 4ac when a or c is negative. Because the formula subtracts 4ac, a negative 4ac term increases the discriminant. The step table is designed to make that visible.
A third mistake is using the discriminant before the equation is in standard form. If the equation is written as ax² + bx = −c or spread across both sides, move all terms to one side first so a, b, and c match ax² + bx + c = 0.
Limitations and scope
This calculator focuses on the quadratic discriminant. Discriminants also exist for cubic and higher-degree polynomials, but those formulas are different and much more complex.
The calculator is an algebra learning and checking tool. It does not rearrange an equation from arbitrary form, graph the entire function, or prove that a parameterized discriminant is positive or negative for every possible parameter value.
Display values are rounded for readability. When exact algebraic form matters, keep the original coefficients and formula work in your written solution.
The discriminant is Δ = b² − 4ac for a quadratic equation ax² + bx + c = 0. It is the part of the quadratic formula under the square root and it tells you the number and type of roots before you finish solving.
What does a positive discriminant mean?
A positive discriminant means the equation has two distinct real roots. Graphically, the parabola crosses the x-axis in two places. If the discriminant is also a perfect square and the coefficients are integers, those roots are rational.
What does a zero discriminant mean?
A zero discriminant means the equation has one repeated real root. The two branches of the quadratic formula produce the same value, and the parabola touches the x-axis at its vertex.
What does a negative discriminant mean?
A negative discriminant means the equation has two complex conjugate roots and no real roots. The related parabola does not cross the x-axis in the real-number plane.
Can the discriminant calculator find the roots too?
Yes. The calculator shows the root values where appropriate: two real roots, one repeated root, a complex conjugate pair, or a linear root when a = 0. The discriminant remains the main classification output.
How do I know if a quadratic is factorable?
For integer coefficients, a non-negative perfect-square discriminant is a strong cue that the roots are rational and the quadratic may factor neatly over the rationals. If the discriminant is positive but not a perfect square, the real roots are usually irrational.
Why is b negative in the formula sometimes?
b is whatever coefficient appears in front of x after the equation is written as ax² + bx + c = 0. If the term is −5x, then b = −5. The calculator accepts the signed value and shows the substitution so you can check it.
Can I use fractions in the coefficient boxes?
Yes. Simple fractions such as 3/2 or −5/4 are accepted. The calculator converts them to numeric coefficient values before computing b², 4ac, the discriminant, roots, and vertex.
What happens if a = 0?
Then the equation is not quadratic. If b is non-zero, it is a linear equation with one root. If both a and b are zero, there is no variable term to solve as a quadratic or linear equation.
Does the discriminant work for cubic equations?
Cubic and higher-degree polynomial discriminants exist, but they use different formulas. This calculator is intentionally scoped to the quadratic discriminant b² − 4ac.
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