💡 Enter coefficients (a, b, c, d) | Complex roots shown when discriminant < 0
📖 What is an Equation Solver?
An equation solver is a tool that finds the value(s) of the unknown variable(s) that make an equation true.
This calculator supports three common types of polynomial equations:
Linear Equation: ax + b = 0 — one solution
Quadratic Equation: ax² + bx + c = 0 — two solutions (real or complex)
Cubic Equation: ax³ + bx² + cx + d = 0 — up to three solutions
🧮 Standard Formulas
Linear: x = −b / a Quadratic: x = [−b ± √(b² − 4ac)] / 2a Discriminant: Δ = b² − 4ac Cubic: Solved using Cardano's method or numerical root-finding.
📝 Worked Examples
Example 1: Linear Equation — 2x + 6 = 0
Given: a = 2, b = 6 Formula: x = −b / a Calculation: x = −6 / 2 = −3
Example 2: Quadratic Equation — x² − 3x + 2 = 0
Given: a = 1, b = −3, c = 2 Discriminant: Δ = (−3)² − 4(1)(2) = 9 − 8 = 1 Since Δ > 0: Two distinct real roots x₁ = (3 + √1) / 2 = 2 x₂ = (3 − √1) / 2 = 1
Linear: a ≠ 0 required. If a = 0, equation becomes b = 0 (no unique solution).
Quadratic: a ≠ 0 required. If a = 0, it's treated as linear.
Cubic: a ≠ 0 required. If a = 0, it's treated as quadratic.
Cubic solver uses numerical approximation — results are accurate to about 6 decimal places.
Complex roots are shown as a ± bi where i = √(−1).
Repeated roots and irrational roots are shown with rounding.
📚 Sources & References
Authoritative Sources
OpenStax — College Algebra, Chapter 2: Equations and Inequalities
Khan Academy — Quadratic Equations and Functions
Stewart, J. — Precalculus: Mathematics for Calculus
Abramowitz & Stegun — Handbook of Mathematical Functions
Wikipedia — Cubic equation, Cardano's method
❓ Frequently Asked Questions
What is the difference between a linear and a quadratic equation?
A linear equation has the form ax + b = 0 and gives one solution.
A quadratic equation has the form ax² + bx + c = 0 and gives two solutions
(which may be real, repeated, or complex).
What does a negative discriminant mean?
A negative discriminant (Δ < 0) means the quadratic has no real roots.
It has two complex conjugate roots of the form a ± bi, where i = √(−1).
Can a cubic equation have fewer than 3 roots?
A cubic equation always has 3 roots in the complex number system, but some may be repeated
(e.g., (x−1)²(x−2) = 0 has roots 1, 1, 2) or complex.
How accurate is the cubic solver?
The cubic solver uses an iterative numerical method accurate to about 6 decimal places.
For most educational and practical purposes, this is more than sufficient.
Does this calculator show step-by-step solutions?
Yes — for linear and quadratic equations, the step-by-step solution is shown (including discriminant
calculation). For cubic, roots are shown with a brief description of the method.
This Equation Solver is provided for informational and educational purposes only.
While we strive for accuracy, results are rounded and may not always match exact algebraic solutions.
For critical mathematical or engineering applications, always verify with authoritative sources
or a qualified professional. Online Calculator Zone is not responsible for any decisions made based on
these calculations.