Type an equation in one variable and get every solution: exact values with radicals and fractions where they exist, complex roots, general solutions for trig equations and accurate numeric roots when no formula applies, plus the working and a graph of both sides.
Last updated:
Reads as
No "=" sign found, so the calculator solves expression = 0.
Try:
Solutions
Graph of both sides
left sideright sidesolutions
Real solutions are the x-values where the two curves cross.
Step by step
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How to use the Equation Solver
Type the equation with one = sign, for example x^3 - 6x^2 + 11x = 6. Without = the expression is set equal to 0.
Choose the variable in Solve for if the equation has more than one letter.
Adjust the range used for numeric searches and the graph if you need roots far from 0.
Press Solve (or Enter). Results also update as you type.
Read the solutions and steps, then copy them with Copy solutions.
Which method the solver uses
Equation type
Example
Method
Linear
2x + 3 = 11
Move terms, divide by the coefficient: x = 4
Quadratic
x² + 2x + 5 = 0
Factor, or quadratic formula; discriminant −16 gives x = −1 ± 2i
Polynomial (degree 3+)
x³ − 6x² + 11x − 6 = 0
Rational root test and division: (x − 1)(x − 2)(x − 3) = 0
Rational
(x + 1)/(x − 2) = 3
Multiply by the denominator, solve, reject values that make it 0
Radical
√(x + 3) = x − 3
Isolate the root, square both sides, check for extraneous roots
Exponential
2^(x+1) = 8
Take logarithms: x + 1 = ln 8 / ln 2 = 3
Logarithmic
ln(2x − 1) = 3
Exponentiate: 2x − 1 = e³
Trigonometric
sin(2x) = 1/2
Inverse function plus the period: 2x = π/6 + 2πn or 5π/6 + 2πn
Absolute value
|2x − 1| = 5
Split into 2x − 1 = 5 and 2x − 1 = −5
Mixed / transcendental
cos(x) = x
Numeric: sign changes on a fine grid, then bisection
The quadratic formula and the discriminant
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac tells you what to expect before you calculate: D > 0 gives two different real roots, D = 0 gives one repeated real root, and D < 0 gives two complex conjugate roots. If D is a perfect square (and a, b, c are integers), the roots are rational and the quadratic factors over the integers.
Tips for entering equations
Use one = sign. Everything is moved to one side internally, so x^2 = 2x + 3 and x^2 - 2x - 3 = 0 give the same roots.
Write multiplication by brackets or juxtaposition: 2(x+1), 3x, x(x-2).
Trig functions use radians. sin(x) = 1/2 has solutions π/6 + 2πn and 5π/6 + 2πn.
If the graph shows a crossing you expected but the list does not, widen the search range – numeric roots are searched only inside it.
Frequently asked questions
What kinds of equations can it solve?
Linear and polynomial equations of any degree, rational equations with x in a denominator, radical equations such as √(x + 3) = x − 3, exponential and logarithmic equations such as 2^(x+1) = 8 or ln(2x − 1) = 3, absolute-value equations and trig equations such as sin(2x) = 1/2. Equations that mix x inside and outside a function, like cos(x) = x, are solved numerically.
How are polynomial equations solved exactly?
The polynomial is factored over the rational numbers using the rational root theorem and polynomial division. Linear factors give rational roots and quadratic factors are solved with the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, giving radicals or complex roots. Factors of degree 3 or more without rational roots are solved numerically to about 12 significant digits.
What is an extraneous solution?
A value produced by the algebra that does not satisfy the original equation. Squaring both sides of a radical equation or multiplying by an expression containing x can introduce them. The solver substitutes every candidate back into the original equation and rejects values that fail or that make a denominator zero, and tells you which ones it removed.
Why do trig equations have infinitely many solutions?
Sine and cosine repeat every 2π and tangent every π, so if x is a solution then so is x plus any whole number of periods. The solver gives the general solution with n standing for any integer, and lists the specific solutions that fall inside your search range.
What are complex roots?
When the discriminant b² − 4ac of a quadratic is negative, there is no real solution, but there are two complex conjugate roots of the form a ± bi, where i² = −1. For example x² + 2x + 5 = 0 has roots −1 ± 2i. Untick Show complex roots to list only real solutions.
Can it solve for a letter other than x?
Yes. Pick the letter in Solve for. Other letters are treated as constants, so ax + b = c gives x = (c − b)/a, and x² + px + q = 0 gives the quadratic formula in terms of p and q. Numeric root finding and the graph need all other letters to be numbers.
How accurate are numeric roots?
Numeric roots are refined by bisection or Durand–Kerner iteration to near the limit of double-precision arithmetic, about 12 or more significant digits. The search scans the chosen x range, so a root outside the range will not be found until you widen it.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.