Equation Solver

Type a linear, quadratic or cubic equation the way you would write it — brackets, terms on both sides, any letter as the unknown. The solver recognises the degree and gives the exact solutions (real or complex), the working step by step, and the graph with its roots, vertex or turning points. Drag the vertex or a root to explore.

AlgebraQuadraticsCubicsGraphing

Inputs

Input examples

Write powers as x^2 or x², x^3 or x³, and multiplication as 2x or 2*x. Both sides of the “=” may contain terms and brackets, such as x(x − 2)(x + 1) = 4; without an “=” the expression is set equal to 0. Fractions and decimals are fine. For several unknowns at once, use the System of equations tab.

Equation

Quadratic and Cubic Equations

A quadratic equation contains the unknown squared. Its graph is a parabola, and the solutions are where it crosses the horizontal axis: two, one or none. The discriminant D = b² − 4ac tells you which before you solve anything; when D is negative the solutions are complex numbers.

A cubic contains the unknown to the third power. Its graph runs from one side of the axis to the other, so it always has at least one real solution, and at most three. By hand you usually guess one root (whole numbers that divide the constant term are good candidates), divide it out and solve the quadratic that is left; when no simple root exists, Cardano’s formula from 1545 still finds them all.

D > 0quadratic: two real solutions

D < 0quadratic: two complex

cubicalways ≥ 1 real root

≤ 2cubic turning points

Live Graph

drag the vertex

y = left side − right side

solutions

vertex — drag it

Result

Type an equation above

Quadratic Formulas

Quantity	Formula

Worked Techniques

Factorising

Look for two numbers that multiply to c and add up to b. A product is zero only when one of its factors is zero, so each bracket gives a solution. Quick — when it works.

Example:

The Quadratic Formula

Always works, for every quadratic; completing the square is how it is derived, and gives the vertex for free. Simplify the square root by taking out square factors: √68 = 2√17.

Example:

A Cubic: Guess a Root, Divide It Out

With whole-number coefficients, every fraction that solves the equation has a numerator that divides d and a denominator that divides a. Divide out the root you find with synthetic division and solve the quadratic that is left. Without a simple root, Cardano’s method takes over.

Example:

Sources & Verification

References

J. Stewart, L. Redlin & S. Watson, Precalculus: Mathematics for Calculus, 7th ed., Cengage, 2016 — Sec. 1.5 (Equations), Sec. 3.1 (Quadratic Functions and Models), Sec. 3.4 (Real Zeros of Polynomials) and Sec. 3.5 (Complex Zeros).
M. Abramowitz & I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1964 — Sec. 3.8 (Algebraic Equations, including 3.8.2 Solution of Cubic Equations).