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Algebra · Polynomial equations

Quadratic & Cubic Equation Solver

Solve second- and third-degree polynomial equations, classify real and complex roots, inspect the discriminant, and verify the equation structure.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Real-coefficient polynomial equations of degree two or three

Quadratic model

ax² + bx + c = 0Δ = b² − 4ac

Quadratic roots

x = (−b ± √Δ)/(2a)

Cubic model

ax³ + bx² + cx + d = 0
Polynomial equations

Find and classify every root

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

ax² + bx + c = 0

Choose the degree and enter every coefficient.

Using the model

Symbols, assumptions and limitations

Interpretation notes

Coefficients are real and the leading coefficient must be non-zero. Quadratic roots use the discriminant directly. Cubic roots are approximated numerically in the complex plane, so displayed decimals are rounded even when internal iteration uses greater precision.

Worked examples

See the method in practice

01

Two real quadratic roots

x² − 5x + 6 = 0 has discriminant 1 and factors as (x − 2)(x − 3), giving roots 2 and 3.

02

Three cubic roots

x³ − 6x² + 11x − 6 = 0 has the three real roots 1, 2, and 3.

Questions

Frequently asked

Can the calculator show complex roots?

Yes. Non-real roots are displayed in a + bi form and occur in conjugate pairs for real coefficients.

Why must the leading coefficient be non-zero?

A zero leading coefficient lowers the degree, so the expression is no longer the selected quadratic or cubic equation.

Are cubic roots exact?

They are found numerically and displayed to at most three decimal places; simple integer roots may still appear exact.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Algebra and Trigonometry — Quadratic Equations

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology