liminfoEquation Solver
Free web tool: Equation Solver
Equation Solver
Solve linear, quadratic equations, or systems of 2 linear equations.
About Equation Solver
The Equation Solver handles three categories of algebraic equations in a single tool. In Linear mode, enter coefficients a and b to solve ax + b = 0 for x. In Quadratic mode, enter a, b, and c to solve ax² + bx + c = 0, with the solver showing both real roots, the double root, or complex conjugate roots depending on the discriminant. In System mode, enter the coefficients of two linear equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) and the solver uses Cramer's rule to find x and y.
The quadratic solver computes the discriminant (b² − 4ac) and branches accordingly: two distinct real roots when the discriminant is positive, a repeated real root when it equals zero, and a pair of complex conjugate roots in the form r ± si when the discriminant is negative. The 2×2 system solver computes the matrix determinant (a₁b₂ − a₂b₁) and applies Cramer's rule; if the determinant is zero, the system has no unique solution. All numerical results are formatted to up to six significant decimal places with trailing zeros removed.
All calculations run entirely in your browser. The interface uses tabbed mode switching so you can move between linear, quadratic, and system solvers without losing your input values. The tool is free, requires no account, and works on any device.
Key Features
- Three solver modes in one tool: linear (ax + b = 0), quadratic (ax² + bx + c = 0), and 2×2 system of linear equations
- Quadratic solver handles real roots (two distinct or double), and complex conjugate roots with imaginary part
- Displays the discriminant for quadratic equations and the matrix determinant for linear systems
- 2×2 system solver uses Cramer's rule: x = (c₁b₂ − c₂b₁) / det, y = (a₁c₂ − a₂c₁) / det
- Detects degenerate cases: no solution when a = 0 in linear mode, infinite solutions when both a and b are zero
- Results formatted to 6 decimal places with trailing zeros stripped for clean output
- 100% client-side — no data is ever sent to a server
- No sign-up, no download, and completely free to use
Frequently Asked Questions
How do I solve a linear equation like 3x − 9 = 0?
Select the Linear tab, enter a = 3 and b = -9. The solver computes x = −b / a = 9 / 3 = 3. The result is displayed instantly as x = 3.
How do I solve a quadratic equation like x² − 5x + 6 = 0?
Select the Quadratic tab, enter a = 1, b = -5, c = 6. The solver computes the discriminant (25 − 24 = 1 > 0), so there are two real roots: x₁ = (5 + 1) / 2 = 3 and x₂ = (5 − 1) / 2 = 2.
What happens when the quadratic discriminant is negative?
When b² − 4ac < 0, the equation has no real solutions. The solver returns complex conjugate roots in the form x₁ = r + si and x₂ = r − si, where r = −b / (2a) is the real part and s = √(−discriminant) / (2a) is the imaginary part.
How does the 2×2 system solver work?
Enter the coefficients for the two equations: a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The solver computes the determinant det = a₁b₂ − a₂b₁. If det ≠ 0, it uses Cramer's rule: x = (c₁b₂ − c₂b₁) / det and y = (a₁c₂ − a₂c₁) / det.
What does "No unique solution (determinant = 0)" mean?
When the determinant of the 2×2 coefficient matrix is zero, the two equations are either parallel (no solution) or identical (infinite solutions). The solver cannot distinguish between these two sub-cases, so it reports that no unique solution exists.
Can I solve equations with decimal or fractional coefficients?
Yes. All input fields accept any numeric value including decimals. For example, you can enter a = 0.5, b = -1.5, c = 0.75 for a quadratic equation with fractional coefficients. Results are computed in floating-point and shown with up to 6 decimal places.
What is a double root in a quadratic equation?
A double root (also called a repeated root) occurs when the discriminant equals zero. In this case, both solutions are identical: x = −b / (2a). Geometrically, the parabola y = ax² + bx + c touches the x-axis at exactly one point.
Is this tool suitable for students learning algebra?
Yes. The tool clearly labels each input field with the coefficient names (a, b, c), shows the discriminant for quadratic equations and the determinant for systems, and distinguishes between real, complex, and degenerate cases with descriptive labels. It is a good companion for algebra coursework.