About Equation Solver
This equation solver is free, needs no sign-up and runs in your browser, so your equations are never uploaded. Type an equation in one variable such as 2(x + 1) = 8, x/2 + x/3 = 5, (x + 1)/(x - 2) = 3 or x^2 - 5x + 6 = 0 and it solves it and writes out the steps in typeset math. Switch to the system modes to solve two or three linear equations together.
For a quadratic it shows the coefficients, the discriminant and what the discriminant says about the roots, then applies the quadratic formula. Roots that are whole numbers or fractions come out exactly, and complex roots are shown in the form a + bi. Irrational roots are kept exact as square roots, such as x = (1 ± √5) ÷ 2. Only roots with no short exact form, such as those of some cubics and higher equations, are shown as decimals marked with the approximately-equal sign.
Cubic and higher equations first look for a rational root with the rational root theorem. If one exists the polynomial is divided by that factor and the smaller equation is solved the same way. If none exists, the remaining roots are found numerically. A graph of the left side minus the right side marks the real roots where it crosses the x-axis.
How to use Equation Solver
- Pick a type: Linear, Quadratic, Cubic, Polynomial roots, 2 equations, 3 equations or Discriminant.
- For a single equation, type it in the box using ^ for powers, such as x^3 - 6x^2 + 11x - 6 = 0. The type is detected from what you enter.
- For a system, fill the grid with the coefficients and the constant of each equation, in the order x, y, z, constant.
- Read the roots or the solution at the top of the result.
- Open the steps to follow the working, and look at the graph to see where the curve meets the axis.
Features
- Linear, quadratic, cubic and higher-degree polynomial equations in one variable, with the degree detected automatically.
- Quadratic roots with the discriminant shown and explained, including complex roots in the form a + bi.
- Exact rational roots shown as integers or fractions, such as x = 1/2, and exact radical and complex roots such as x = 1 ± √2 or x = −1 ± 2i.
- Cubic and higher equations solved by the rational root theorem, reducing the degree step by step, with a numeric fallback when no rational root exists.
- Cubic discriminant reported, so you know whether to expect three real roots or one real and two complex.
- Systems of 2 and 3 linear equations solved by Gauss-Jordan elimination, with the augmented matrix and each row operation typeset.
- Detects systems with no solution, and gives a parametric solution such as x = 2 − t, y = t for systems with infinitely many.
- Brackets, fractions and rational equations through the expression engine; values that make a denominator zero are excluded.
- A graph of the left side minus the right side with the real roots marked, which you can zoom, pan and export.
- A standalone discriminant mode for b² − 4ac.
The quadratic formula and the discriminant
For ax² + bx + c = 0 with a ≠ 0, the roots are given by the quadratic formula. The part under the square root, D = b² − 4ac, is the discriminant, and its sign decides the type of roots.
If D is positive there are two different real roots. If D is zero there is one repeated real root, x = −b ÷ 2a. If D is negative there are no real roots and the two roots are complex conjugates, x = (−b ± i√(−D)) ÷ 2a.
x = (−b ± √(b² − 4ac)) / 2a
D = b² − 4acWorked examples
x^2 - 5x + 6 = 0 gives x = 2 or x = 3
a = 1, b = −5, c = 6. D = 25 − 24 = 1, so √D = 1 and x = (5 ± 1) ÷ 2.
x^2 + 2x + 5 = 0 gives x = −1 + 2i and x = −1 − 2i
D = 4 − 20 = −16 is negative, so the roots are complex: x = (−2 ± 4i) ÷ 2.
x^3 - 6x^2 + 11x - 6 = 0 gives x = 1, x = 2, x = 3
x = 1 is a rational root. Dividing by (x − 1) leaves x² − 5x + 6 = 0, which has roots 2 and 3.
2x + y = 5 and x − y = 1 gives x = 2, y = 1
Gauss-Jordan elimination reduces the augmented matrix until the left side is the identity, leaving the solution in the last column.
What the discriminant tells you
| Discriminant D = b² − 4ac | Roots | Example |
|---|---|---|
| D > 0 | Two different real roots | x² − 5x + 6 = 0, D = 1 |
| D = 0 | One repeated real root | x² − 4x + 4 = 0, D = 0, x = 2 |
| D < 0 | Two complex conjugate roots, no real roots | x² − 4x + 5 = 0, D = −4 |
Good to know
How cubic and higher equations are solved here
The rational root theorem says that any rational root of a polynomial with integer coefficients has a numerator that divides the constant term and a denominator that divides the leading coefficient. The solver tests those candidates, divides out the first root it finds and repeats on the quotient until a quadratic is left, which it solves exactly.
Many polynomials have no rational root. x³ − 2x − 5 = 0 is one: its one real root is about 2.094551482, and the other two roots are the complex pair −1.047275741 ± 1.135939889i. For those, the solver uses Durand-Kerner iteration, so the roots are accurate decimals rather than closed forms. Equations of degree five and higher have no general solution in radicals, so numeric roots are the usual answer there too.
Systems of equations
A system is written as an augmented matrix with one row per equation and the constants in the last column. Gauss-Jordan elimination uses row operations to reach reduced row echelon form. A row that reads 0 = 7 means the equations contradict each other and there is no solution. Fewer pivots than unknowns means some variables are free, so there are infinitely many solutions.
Privacy: how it works
- Equations are solved by code running in your browser tab. They are not sent to Ulkaa or anyone else, and there is nothing to sign in to.
Tips
- Brackets, fractions and rational equations are accepted, so (x + 1)(x + 2) = 0, x/2 + 1 = 4 and (x + 1)/(x - 2) = 3 can be typed as they are.
- Move nothing yourself: write the equation as you have it, with terms on both sides if needed, and the solver collects them.
- Use a decimal or a fraction for coefficients, such as 0.5x + 2 = 5 or 1/2x + 2 = 5.
- In the system grid, leave a cell blank for a zero coefficient.
- For one variable, the equation can use x or any single letter, but only one letter per equation.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your equations are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your equations are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I solve a quadratic equation?
Write it as ax² + bx + c = 0 and put a, b and c into the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. The solver does this and lists each step.
What is the discriminant?
It is b² − 4ac, the expression under the square root in the quadratic formula. Positive means two real roots, zero means one repeated root, negative means two complex roots.
How do I solve simultaneous equations?
Use the 2 equations or 3 equations mode and enter the coefficients. The solver row-reduces the augmented matrix and reads off the answer, or tells you if there is none or infinitely many.
What if there are no real roots?
Then the roots are complex numbers. A quadratic with a negative discriminant gives a pair a ± bi, and the graph does not touch the x-axis.
Can it show steps?
Yes. Every mode lists its working: the coefficients, the discriminant, the substitution, the division by a found root, or the row operations.
Are the answers exact?
Whole-number and fractional roots are exact. Irrational quadratic roots stay exact as square roots. Only roots with no short exact form, such as those of a cubic with no rational root, are decimals to about ten places and are marked as approximate.
Can I type brackets such as (x + 2)(x − 3) = 0?
Yes. The equation is read by a full expression parser, so brackets, fractions like x/2 and rational equations all work without expanding anything first.
Popular Equation Solver pages
- Quadratic Equation SolverType a quadratic and get its roots, the discriminant and the working.
- Quadratic Formula CalculatorSee the quadratic formula applied to your numbers, line by line.
- Linear Equation SolverSolve for x in an equation with x on one or both sides.
- System of Equations CalculatorEnter the coefficients of two equations and get x and y with each row operation.
- 3 Variable System of Equations SolverSolve three equations in x, y and z and see every row operation.
- Simultaneous Equations SolverSolve two equations at once and check the answer against both.
- Cubic Equation SolverFind all three roots of a cubic, real and complex.
- Polynomial Roots CalculatorFind every root of a polynomial of any degree and plot the real ones.
- Discriminant CalculatorEnter a, b and c to get b² − 4ac and the type of roots.