Equation Solver

What does the equation solver do?

It reads an equation in one unknown the way you type it and finds its roots. If the equation is a polynomial it is solved by degree: a linear equation is rearranged for the unknown, a quadratic is solved with the discriminant and the quadratic formula, a cubic is classified with the Cardano discriminant, and higher degrees (up to 12) have all their real and complex roots found numerically. For equations that are not polynomials (such as sin(x) = x/2) it searches the real roots inside the interval you give. The second tab solves a linear system in 2, 3 or 4 unknowns by Gaussian elimination, step by step.

The quadratic formula

For a·x² + b·x + c = 0 the discriminant is Δ = b² − 4ac. If Δ is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are two complex conjugate roots. The roots are x = (−b ± √Δ) / (2a).

Example

Type x^2 - 5x + 6 = 0. With a = 1, b = −5 and c = 6 the discriminant is Δ = 25 − 24 = 1. The roots are (5 − 1)/2 = 2 and (5 + 1)/2 = 3. Check by substitution: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.

Exact and approximate results

When the coefficients are whole numbers, fractions or decimals the work is done with fractions. Rational roots are given exactly, such as 3/2, quadratic roots with radicals are written like (1 + √5)/2, and the decimal value is shown next to them. Repeated roots are separated with exact arithmetic first, so no accuracy is lost and the multiplicity of each root is shown. If the coefficients contain numbers that are not rational, such as π or a square root, the result is given in decimals only.

Systems of equations

Enter the coefficients in the table. The tool reduces the augmented matrix with row operations and shows every step. If there is a unique solution the values are given as fractions and decimals. If the system has infinitely many solutions or none, the tool says so.

Limits

For equations that are not polynomials only real roots inside the given interval are searched: the interval is split into 4000 parts and bisection is applied wherever the sign changes. Roots outside the interval or very close together may be missed, so narrow or widen the interval yourself. Trigonometric functions take radians. For cubic and higher degrees no exact form with radicals is given and irrational roots are written as decimals. Inequalities and equations with parameters are not supported.