What types of equations can this solver handle?

This solver handles linear equations (degree 1), quadratic equations (degree 2), cubic equations (degree 3), and general polynomial equations in one variable.

How do I enter a quadratic equation?

Type the equation using standard notation. For example: x^2 - 5x + 6 = 0 or 2x^2 + 3x - 2 = 0. The solver automatically detects the degree and applies the correct method.

What is the discriminant?

The discriminant is b² - 4ac for a quadratic ax² + bx + c = 0. If it is positive, the equation has two distinct real roots. If zero, one repeated root. If negative, two complex roots with no real solutions.

Can this solver handle equations not equal to zero?

Yes. You can enter equations like x^2 = 9 or 2x + 3 = 7. The solver rearranges to standard form automatically before solving.

Does the solver show working steps?

Yes. For every equation type, the solver shows the full algebraic working: standard form, discriminant calculation, quadratic formula substitution, factoring steps, and verification by substituting the roots back.

Discriminant Calculator — Step-by-Step Linear, Quadratic & Polynomial

How the Discriminant Calculator Works

Enter any equation in one variable using standard notation. The solver detects the type — linear, quadratic, or cubic — and applies the correct algebraic method. Every solution includes full step-by-step working and a verification check.

Linear equations (degree 1)

Example: 2x + 3 = 7. The solver collects variable terms, moves constants to the other side, then divides by the coefficient to isolate x. Result: x = 2.

Quadratic equations (degree 2)

Example: x² − 5x + 6 = 0. The solver calculates the discriminant Δ = b² − 4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated root. If Δ < 0: no real roots. The quadratic formula x = (−b ± √Δ) / 2a is applied and simplified.

Cubic equations (degree 3)

Example: x³ − 6x² + 11x − 6 = 0. The solver attempts factoring using the rational root theorem, then identifies all three roots and verifies each one.

Entering equations

You can enter equations with or without the right-hand side. x^2 - 4 = 0 and x^2 - 4 are both valid — the solver assumes = 0 if no equals sign is present. Use ^ for powers.

Frequently Asked Questions

The discriminant controls the entire behavior of a quadratic equation. Before calculating roots, the solver evaluates b² − 4ac to determine whether the parabola intersects the x-axis twice, touches it once, or never reaches it at all. This allows the calculator to classify the equation instantly before applying solving methods and graph rendering.

The calculator extracts the coefficients automatically, computes the discriminant, and uses the result to determine root behavior, graph structure, verification logic, and solving flow. Positive discriminants trigger two real roots, zero produces a repeated root, and negative values indicate complex solutions and a parabola that never crosses the x-axis.

A negative discriminant means the quadratic equation has no real x-intercepts. On the graph, the parabola stays completely above or below the x-axis depending on the leading coefficient. The calculator reflects this visually in the graph while also explaining the result in the diagnostics panel.

Yes. The discriminant calculator works independently of factorability. Even if an equation produces irrational or complex roots, the solver still calculates the discriminant accurately, classifies the root type, and generates full step-by-step analysis automatically.

Yes. Every solution includes discriminant interpretation alongside the equation’s roots, graph behavior, and diagnostics. This helps explain not only what the answer is, but why the quadratic behaves that way mathematically.

Yes. The calculator can generate a clean downloadable PDF containing the equation, discriminant calculation, diagnostics, graph visualization, and full step-by-step breakdown for study, revision, or printing.