Algebra Calculator

Solve a linear or quadratic equation for x, a system of two simultaneous equations for x and y, find every root of a higher-degree polynomial, or divide two polynomials — with the formula worked out step by step, and a graph underneath where one applies.

ax + b = cx + d

a b c d

How to use this calculator

Choose a mode above, then fill in the coefficients - results and the worked formula appear underneath as soon as every field has a value.

Linear Solves ax + b = cx + d for x - the general two-sided form, not pre-simplified, so you can enter a problem the way it's usually written. The graph shows both sides as lines, crossing at the solution.
Quadratic Solves ax² + bx + c = 0 using the discriminant and the quadratic formula. Two real roots, one repeated root, or two complex roots, depending on the discriminant's sign - all three cases are handled, not just the friendly one.
Simultaneous Solves two linear equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) for x and y using Cramer's rule. Parallel lines correctly report no solution, and identical lines correctly report infinitely many.
Complex roots When a quadratic's discriminant is negative, the parabola never crosses the x-axis - the roots are shown as a + bi instead of reporting "no solution", since they do exist, just not as real numbers.
Polynomial Finds every root (real and complex) of a degree 3-6 polynomial using a numerical method (Durand-Kerner) rather than a closed-form formula - degree 5 and up famously has no general formula at all (Abel-Ruffini theorem), so a numerical method is used uniformly across every degree here rather than hand-coding the cubic and quartic formulas as one-off special cases.
Long Division Divides one polynomial by another of a lower degree, showing the quotient, remainder, and each step's "bring down the next term" working - the same algorithm as dividing numbers by hand, just with x's instead of digits.