What types of equations can this equation solver handle?

This equation solver is built for core algebra problems. It can solve linear equations in one variable (ax + b = c), quadratic equations (ax^2 + bx + c = 0) with step-by-step solutions, and linear systems of 2×2 or 3×3 equations. For more general equations of the form f(x) = 0, you can use the numeric solver, which approximates a real root using the Newton–Raphson method.

How accurate is the numeric equation solver?

The numeric solver uses a Newton–Raphson root-finding method with a default tolerance of 1×10^-7 and a limited number of iterations. For well-behaved functions and a reasonable initial guess, this typically provides at least 6–7 correct decimal digits. However, convergence is not guaranteed: if the derivative is very small, the function is highly oscillatory, or the initial guess is far from any root, the method may fail or converge slowly.

Can this solver find complex solutions?

For quadratic equations, the solver can detect when the discriminant is negative and report that the solutions are complex, including their exact form with the imaginary unit i. For linear equations and numerical solving, only real-valued solutions are supported. If you need full complex arithmetic or symbolic algebra, a dedicated computer algebra system may be more appropriate.

How should I format equations for the solver?

For the analytic linear and quadratic solvers, use forms like 2x + 3 = 7 or 3x^2 - 5x + 2 = 0. Use x as the single variable and the caret ^ for exponents. For the numeric solver, you can type general expressions such as sin(x) - x/2 = 0 or e^x - 3 = 0, with standard functions sin, cos, tan, log, ln, sqrt, and exp. Avoid using semicolons or programming syntax—stick to mathematical notation.

Equation Solver

Solve linear and quadratic equations, or small systems of equations, with step-by-step explanations and a numeric root finder for general equations of the form f(x) = 0.

Equation in one variable (x)

Use x as the variable and ^ for powers (e.g., x^2). You can also enter equations like sin(x) - x/2 = 0 for the numeric solver.

Solving method

Initial guess (numeric mode)

Used only for the numeric solver.

Tolerance

Smaller values require more iterations.

The solution and step-by-step explanation will appear here.

Solve a small linear system in the form Ax = b. Enter the coefficients and constants for each equation.

System size

Equation	x	y	z	=	Constant
1				=
2				=
3				=

For a 2×2 system, only the first two rows and the x, y columns are used. The third row and z column are ignored.

The solution and intermediate matrix steps will appear here.

How this equation solver works

This tool is designed for core algebra and pre-calculus work. It combines exact algebraic formulas for linear and quadratic equations with a numeric solver for more general problems, plus a compact linear system solver for small systems.

1. Linear equations in one variable

A linear equation in one variable has the form ax + b = c, where a, b, and c are real numbers and a ≠ 0.

Solving a linear equation:

Start from: ax + b = c

Subtract b from both sides: ax = c - b

Divide by a: x = (c - b) / a

2. Quadratic equations

A quadratic equation has the form ax^2 + bx + c = 0 with a ≠ 0. The solutions are given by the quadratic formula.

Quadratic formula:

Given ax^2 + bx + c = 0, define the discriminant D = b^2 - 4ac.

If D > 0, there are two distinct real roots.
If D = 0, there is one repeated real root.
If D < 0, the roots are complex conjugates.

The roots are: x = (-b ± √D) / (2a)

3. Numeric solver (Newton–Raphson)

For equations that cannot be easily solved symbolically, we treat the problem as finding a root of a function f(x) = 0. The numeric solver uses the Newton–Raphson iteration:

x_{n+1} = x_n - f(x_n) / f'(x_n)

The derivative f'(x) is approximated numerically, and the process continues until the change between iterations is smaller than the specified tolerance or the maximum number of iterations is reached.

4. Systems of linear equations

For systems, we represent the equations as a matrix equation Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constants vector. The solver uses a simple Gaussian elimination algorithm with partial pivoting for stability.

Example (2 × 2 system):

a₁₁x + a₁₂y = b₁

a₂₁x + a₂₂y = b₂

Under the condition that the determinant det(A) = a₁₁a₂₂ - a₁₂a₂₁ ≠ 0, the system has a unique solution.

Tips for reliable results

Use consistent notation: x as the unknown, ^ for powers, and parentheses for grouping.
For the numeric solver, choose an initial guess close to where you expect a root.
If the solver reports that the system is singular or nearly singular, your equations may be dependent or inconsistent.