Which equations can be solved exactly?

The tool provides exact, step-by-step solutions for linear and quadratic equations in one variable. More complex equations are solved numerically.

Can it solve systems of equations?

Yes. It solves 2×2, 3×3, and 4×4 linear systems using Gaussian elimination with partial pivoting.

How accurate are numeric solutions?

Numeric solutions use bracketing and Newton refinement with configurable tolerance. Residuals and convergence indicators are reported.

Does it support functions like sin and log?

Yes. The numeric solver supports common functions including sin, cos, tan, log, ln, exp, sqrt, and abs.

Is the tool accessible?

The interface follows WCAG 2.1 AA with full keyboard navigation, visible focus states, ARIA labeling, and responsive design.

Equation Solver

This professional-grade equation solver helps students, educators, and engineers solve algebraic problems faster and with confidence. It provides exact step-by-step solutions for linear and quadratic equations, robust numeric roots for complex equations, and a reliable solver for 2×2 to 4×4 linear systems—designed mobile-first and fully accessible.

Single Equation Linear System

Equation *

Variable name *

Method *

Numeric search interval

Min

Max

Tolerance

Max iterations

Results

Status Awaiting input

Data Source and Methodology

Authoritative references:

NIST Digital Library of Mathematical Functions (DLMF), Release 1.2.1, 2024. https://dlmf.nist.gov/
R. L. Burden & J. D. Faires, Numerical Analysis, 10th ed., Cengage Learning, 2016.

All computations strictly follow formulas and methods from these sources. All calculations are rigorously based on the formulas and data provided by this source.

The Formulas Explained

Linear equation: $$ ax + b = 0 \quad \Rightarrow \quad x = -\frac{b}{a}, \quad a \neq 0. $$

Quadratic equation: $$ ax^2 + bx + c = 0 \;\Rightarrow\; x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\ a \neq 0. $$

Bisection method (numeric): $$ \text{Given } [L,R] \text{ with } f(L)f(R) \lt 0,\ m = \frac{L+R}{2},\ \text{choose the subinterval where the sign changes, repeat.} $$

Newton refinement: $$ x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}, \quad f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}. $$

Glossary of Variables and Outputs

Equation: The expression of equality to solve, e.g., 2x^2 - 5x + 2 = 0.
Variable: The unknown symbol to solve for (default: x).
Method: Auto, Symbolic Only (up to quadratic), or Numeric Only.
Min/Max: Interval scanned for numeric roots.
Tolerance: Target precision for numeric methods.
Max iterations: Safety cap for numeric convergence loops.
Solutions: Exact values (when symbolic) or numeric approximations.
Residual: Maximum |f(x)| across reported solutions (smaller is better).

Worked Example

How It Works: A Step-by-Step Example

Problem: Solve 2x + 5 = 19.

Rewrite: 2x + 5 = 19 → 2x = 19 − 5 = 14.
Divide both sides by 2: x = 14 ÷ 2 = 7.
Result: x = 7.

Using the quadratic form, for example x^2 − 5x + 6 = 0, the solver applies:

$a=1,\ b=-5,\ c=6,\ \Delta=b^2-4ac=25-24=1 \Rightarrow x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{5\pm1}{2}=\{3,2\}.$

Frequently Asked Questions (FAQ)

What types of equations can this solve exactly?

Linear and quadratic single-variable equations are solved symbolically with steps. Cubic and higher degrees, or equations with trigonometric/exponential functions, are solved numerically.

How are numeric roots found?

The solver scans the given interval for sign changes, applies bisection for bracketing, and refines with Newton’s method using a numerically estimated derivative.

How do I write functions and powers?

Use sin(x), cos(x), tan(x), ln(x) or log(x), exp(x), sqrt(x), abs(x). Use ^ for powers and include multiplication explicitly (e.g., 2*x, not 2x).

Can I solve systems of equations?

Yes, use the Linear System mode to solve 2×2, 3×3, or 4×4 systems via Gaussian elimination (with partial pivoting) for numerical stability.

How do I interpret residuals?

The residual |f(x)| indicates how well a value satisfies the equation. Values near or below the tolerance indicate high accuracy.

Does the tool support complex solutions?

For quadratics, complex solutions are displayed when the discriminant is negative. Numeric solving currently targets real roots within the interval.

Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Board.
Last reviewed for accuracy on: September 14, 2025.