The Discriminant Calculator provided by Hesapstan computes and interprets the discriminant for quadratic equations ax²+bx+c and cubic equations ax³+bx²+cx+d. It shows the formula, the substitution step, the numeric discriminant value, and the root-type interpretation. It does not solve for the actual roots or factor the polynomial.
The discriminant tells you the type of roots, not the roots themselves
A discriminant is a special expression that gives information about the roots of a polynomial equation. For a quadratic equation, D=b²−4ac tells whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots.
This calculator supports two modes: quadratic and cubic. Each mode uses its own formula. The output is the discriminant value and its interpretation; it is not a full root solver.
This version supports quadratic and cubic equations only. Quartic and higher-degree polynomials, actual root solving, factoring, and graph output are not included.
This calculator shows the formula, substitution, value, and interpretation
In quadratic mode you enter a, b, and c. In cubic mode you enter a, b, c, and d. In both modes, a cannot be zero because the equation would no longer have the selected degree.
- Choose the equation degree: quadratic or cubic.
- Enter the numeric coefficients, including negative or decimal values if needed.
- Read the symbolic discriminant formula for that degree.
- Check the substitution line showing your coefficients inside the formula.
- Use the numeric value and interpretation tag to understand the root type.
The discriminant determines the type of roots. This calculator does not calculate the actual root values.
The quadratic discriminant is D=b²−4ac
For ax²+bx+c=0, the discriminant is D=b²−4ac. It is closely tied to the quadratic graph: it tells how the parabola relates to the x-axis.
- D>0 means two distinct real roots; the graph crosses the x-axis twice.
- D=0 means one repeated real root; the graph touches the x-axis once.
- D<0 means no real roots; the equation has two complex conjugate roots.
For a quadratic equation, D<0 means there are no real roots. Over the complex numbers, the equation still has two complex conjugate roots.
The cubic discriminant uses a longer formula for root character
For ax³+bx²+cx+d=0, the discriminant is Δ=18abcd−4b³d+b²c²−4ac³−27a²d². The formula is less familiar than the quadratic discriminant, but its role is similar: it describes the character of the roots.
- Δ>0 means three distinct real roots.
- Δ=0 means at least one repeated root; exact multiplicity requires more analysis.
- Δ<0 means one real root and two non-real complex conjugate roots.
The cubic discriminant is a more advanced topic than the standard quadratic discriminant. This calculator shows the formula and interpretation, but it does not factor the cubic or solve its roots.
Worked examples show how substitution changes the result
Quadratic example: for x²−5x+6=0, a=1, b=−5, and c=6. D=b²−4ac = (−5)²−4·1·6 = 25−24 = 1. Since D>0, the quadratic has two distinct real roots.
Cubic example: for x³−6x²+11x−6=0, a=1, b=−6, c=11, and d=−6. Substituting these into Δ=18abcd−4b³d+b²c²−4ac³−27a²d² gives Δ=4. Since Δ>0, the cubic has three distinct real roots.
The examples show how to compute and interpret the discriminant. They do not turn this calculator into a root solver.
Discriminant, determinant, and root solving are different ideas
The discriminant describes the root structure of a polynomial. A determinant is a matrix concept. They sound similar, but this calculator is not a determinant calculator.
- A discriminant tells the type of roots; it does not list the root values.
- For a quadratic, D<0 means complex conjugate roots, not no answer at all.
- For a cubic, Δ=0 means at least one repeated root, but not the full multiplicity breakdown.
- Quartic and higher-degree discriminants are outside this version's scope.
A negative coefficient inside a power or product can change the discriminant substantially. The substitution line helps you catch sign mistakes.
The discriminant can guide your next calculation
After the discriminant tells you the root type, you may need another tool to continue. For a quadratic equation, completing the square can show numerical roots and vertex form, while factoring may be useful when simple factors exist.
- Use this discriminant calculator to identify the root type quickly.
- Use completing the square when you need step-by-step quadratic roots.
- Use reverse FOIL or factoring when the quadratic has convenient integer factors.
- Review complex conjugates when a negative quadratic discriminant leads to non-real roots.
Frequently Asked Questions
What is the discriminant?
The discriminant is an expression that indicates the type of roots of an equation. For a quadratic, it is D=b²−4ac.
Does this calculator find the roots?
No. It calculates the discriminant and interprets the root type. It does not output the actual root values.
What does D<0 mean?
For a quadratic, D<0 means there are no real roots, but there are two complex conjugate roots.
What does D=0 mean?
For a quadratic, D=0 means one repeated real root. For a cubic, Δ=0 means at least one repeated root, but exact multiplicity needs additional analysis.
What is the cubic discriminant formula?
For ax³+bx²+cx+d=0, the formula is Δ=18abcd−4b³d+b²c²−4ac³−27a²d².
Are quartic equations supported?
No. This version supports only quadratic and cubic discriminants.
Is discriminant the same as determinant?
No. A discriminant is about polynomial roots; a determinant is a matrix quantity.
Why is the cubic formula longer?
A cubic has more coefficient interactions and more possible root configurations, so its discriminant expression is more complex than b²−4ac.