Online math calculators

Solve linear, quadratic and cubic equations, work through linear systems with Cramer’s rule and plot the graph of functions. Pick a tool from the menu on the left or from the cards below; formulas, notes and frequently asked questions are further down.

Featured tools

CalculatorQuadratic equationsax² + bx + c = 0, with the discriminant Δ = b² − 4ac. CalculatorCramer’s ruleLinear systems: x = D_x / D, if D ≠ 0. CalculatorFunction plotterPlot the graph of a function y = f(x).

Equations

Three solvers for the most common polynomial equations. Choose the degree: the tool opens at the top of this page.

CalculatorLinear equationsax + b = 0, with solution x = −b / a (a ≠ 0). CalculatorQuadratic equationsax² + bx + c = 0, with the discriminant Δ = b² − 4ac. CalculatorCubic equationsax³ + bx² + cx + d = 0, with at least one real root.

Systems and graphs

CalculatorCramer’s ruleLinear systems: x = D_x / D, if the determinant D ≠ 0. CalculatorFunction plotterPlot the graph of a function y = f(x).

Main formulas

The reference formulas for the problems covered by the tools in this category.

Problem	Form	Solution
Linear equation	ax + b = 0	x = −b / a, with a ≠ 0
Quadratic equation	ax² + bx + c = 0	x = (−b ± √Δ) / 2a, with Δ = b² − 4ac
Cubic equation	ax³ + bx² + cx + d = 0	Cardano’s formula or numerical methods
2 × 2 system (Cramer)	a₁x + b₁y = c₁, a₂x + b₂y = c₂	x = D_x / D, y = D_y / D, with D = a₁b₂ − a₂b₁ ≠ 0

Reference formulas for linear, quadratic and cubic equations and for Cramer’s rule.

Practical notes

An equation of degree n has n complex roots, counted with multiplicity. For a quadratic equation, the sign of the discriminant Δ tells how many real solutions there are: two if Δ > 0, one repeated if Δ = 0, none if Δ < 0. Cramer’s rule requires the determinant D to be nonzero: if D = 0 the system has no solution or infinitely many. On the graph of a function, the real solutions of f(x) = 0 are the points where the curve meets the x-axis.

Frequently asked questions

How do you solve a linear equation?

You isolate x: from ax + b = 0 you get x = −b / a. Example: 3x + 6 = 0 gives x = −6 / 3 = −2.

How do you solve a quadratic equation?

Compute the discriminant Δ = b² − 4ac and then x = (−b ± √Δ) / 2a. Example: x² − 5x + 6 = 0 has Δ = 25 − 24 = 1, so x = (5 ± 1) / 2, that is x = 3 and x = 2.

How does Cramer’s rule work?

Each unknown is the ratio of one determinant to the determinant D of the coefficients. Example: in the system 2x + y = 5, x − y = 1 we have D = −3, D_x = −6 and D_y = −3, so x = 2 and y = 1.

How many roots does a cubic equation have?

With real coefficients it has at least one real root and at most three. Example: x³ − 6x² + 11x − 6 = 0 has the roots 1, 2 and 3.