# Online Equation Solver - Linear, Quadratic, Cubic and 2×2 Systems

> Free online equation solver: enter coefficients for linear, quadratic (with discriminant, vertex and complex roots) and cubic equations, plus 2×2 systems.

- URL: https://qilingtools.com/en/math/equation
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter the coefficients and get the solution. Quadratics also show the discriminant, real or complex roots, the vertex, the axis of symmetry, the factored form and the sum and product of the roots; systems show the determinant and Cramer’s rule.

## How to use

1. Pick the equation type: linear, quadratic, cubic or a 2×2 system
2. Fill in each coefficient (decimals and negatives are fine)
3. Read the answer and the working, and copy what you need

## Features

- Quadratic: discriminant, real or complex roots, vertex, axis, factored form
- Cubic: all real roots (Cardano plus Newton refinement)
- 2×2 systems solved with Cramer’s rule, with a uniqueness check
- Live results with working you can copy into your homework

## FAQ

### How is the discriminant computed?

Δ = b²-4ac. Δ>0 gives two distinct real roots, Δ=0 a repeated root, and Δ<0 a pair of complex conjugate roots, which are shown as well.

### What if a is 0?

The quadratic degenerates into the linear equation bx+c=0; the tool says so and solves that instead.

### How do I know a system has no unique solution?

Check the determinant D = a₁b₂ - a₂b₁. When D = 0 the equations are proportional, so there is either no solution or infinitely many.

### Can I use decimal coefficients?

Yes — decimals and negatives are supported, and results are given to 10 significant digits.

## Related tools

- [Graphing Calculator](https://qilingtools.com/en/math/graph): Free online function graphing calculator: type sin(x), x^2 or 1/x and see the curve at once, with several functions, parametric and polar curves, PNG export.
- [Scientific Calculator](https://qilingtools.com/en/math/calculator): Free online scientific calculator: type 2*(3+4)^2, sin(pi/6) or log(8,2) and see the answer as you type, with degrees or radians and a history list.
- [Fraction Calculator](https://qilingtools.com/en/math/fraction): Free online fraction calculator: add, subtract, multiply and divide fractions with automatic reduction, plus fraction ↔ decimal with repeating digits.
- [Percentage Calculator](https://qilingtools.com/en/math/percentage): Free online percentage calculator: X% of Y, X as a percent of Y, increase and decrease, a value after a percentage change and the original behind it.
- [Prime Factorization](https://qilingtools.com/en/math/prime): Free online prime factorization: factor an integer with exponents, list every divisor and their sum, test primality with neighbouring primes, or list a range.
- [GCD and LCM](https://qilingtools.com/en/math/gcd-lcm): Free online GCD and LCM calculator: works with two or more integers, says whether they are coprime, shows each prime factorization and every Euclidean step.
- [Statistics Calculator](https://qilingtools.com/en/math/statistics): Free online statistics calculator: paste a data set for the mean, median, mode, range, quartiles, variance, standard deviation, standard error and outliers.
- [Combinations](https://qilingtools.com/en/math/combination): Free online combination and permutation calculator: exact C(n,r), P(n,r) and n! with big integers, plus repetition variants and the total number of subsets.