Free Equation Solver

Worked example

To solve x² - 2x - 2 = 0, type x^2-2x-2=0 into "Type an equation" and press "Solve". The page shows the exact answer x = 1 ± √3, with x₁ ≈ -0.7321 and x₂ ≈ 2.7321 under it, a line "Discriminant = 12" marked "two distinct real roots", and the vertex (1, -3). In "Step by step" the discriminant 12 is simplified as "√12 = 2√3", which is where the √3 comes from. Switch "Show answers as" to "Decimals" and only x₁ = -0.7321 and x₂ = 2.7321 remain.

Which settings to use, and the limits

• Type the equation the way you would write it. In "Type an equation" write x^2-5x+6=0, 2x+3=7 or 3x-2y=5; x+y=4. Powers can be x^2 or x², 2x means 2 times x, and fractions (1/2), decimal commas (0,5x=3), √2, sqrt(2), pi and brackets such as (x+1)(x-2)=0 all work. Separate the equations of a system with a semicolon or a comma. The unknown can be any single letter, and capitals count as small letters. A whole word such as price is refused, so write x*3=45 instead of price*3=45.
• Know what it can and cannot solve. It solves linear, quadratic and cubic equations in one unknown, and linear systems of 2 or 3 equations with 2 or 3 unknowns. You can type at most 3 equations. Powers above 3, inequalities such as x>3, systems with x² or x*y in them, and functions such as sin or log are refused with a message. Text with no = sign, such as x^2-5x+6, is treated as equal to 0.
• Switch between exact answers and decimals. When an exact form exists and differs from the decimals, "Show answers as" offers "Exact" and "Decimals". "Exact" gives fractions such as x = 13/5 and roots such as x = 1 ± √3, with the decimals underneath; "Decimals" rounds to 4 places. Typing a decimal, as in 0,5x+1,25=3, switches to "Decimals" by itself. Exact roots need coefficients that become whole numbers no larger than 2,000 once fractions are cleared. Otherwise you get decimals only: x^2-3000x+2=0 gives x₁ = 0.0007 and x₂ = 2999.9993.
• Use the discriminant, vertex and graph for quadratics. Under the roots a quadratic shows "Discriminant" with its nature ("two distinct real roots", "one repeated real root" or "two complex conjugate roots") and the "Vertex". A negative discriminant gives complex roots such as x = -1 ± 2i. The graph marks the real roots and the vertex. "Download graph as PNG" saves equation-graph.png at 1,044 by 690 pixels, "Copy graph as image" appears only in browsers that allow copying images, and "Open in Graphing Calculator" plots the same polynomial there. Linear and cubic equations get a graph too, but systems do not.
• Cubics and systems show how they were solved. A cubic with a whole-number or simple-fraction root is solved by the rational root test: x^3-6x^2+11x-6=0 gives x₁ = 1, x₂ = 2 and x₃ = 3, and the steps show the quadratic left after dividing out the first root. Without such a root, Cardano's method gives decimals, as in x^3-2=0 with x₁ = 1.2599 and a complex pair. A system is solved by row operations on an augmented matrix, with each step shown. Contradicting equations give "No solution." and dependent ones give "Infinitely many solutions."
• Copy, share or come back to an equation. "Copy" puts the equation, the answers and the steps on the clipboard as text. "Share result" makes a link that reopens the same answer; the equation travels in the part of the address after the #, which is not sent to a server. The last 5 equations you solved are kept in this browser under "Recent equations", and "Clear history" removes them. The symbol row under the box (x, y, ², ^, √, brackets, =, / and a delete key) is for phone keyboards.

Common problems and how to fix them

It says "Could not read that equation. Try something like x^2-5x+6=0, 2x+3=7 or 3x-2y=5; x+y=4."

The text could not be turned into a polynomial. Common causes are a bracket that is not closed, nothing on one side of =, two = signs, no unknown at all (5=5), and an unknown inside a function, a denominator or an exponent: sin(x)=0, 1/x=2, sqrt(x)=2 and 2^x=8 are all refused. Clear the fraction first (1/x=2 becomes 1=2x), and write squares as x^2. Note that x2 is read as 2 times x, not x squared.

It says "Only equations up to the third power (x³) are supported here. For higher powers, plot the curve and read off where it crosses zero."

The equation, after multiplying out brackets, has a power above 3, such as x^4-5x^2+4=0 or (x+1)^4=0. Use the "Open the Graphing Calculator" link under the message to see where the curve crosses zero. If only even powers appear, substitute u = x²: u²-5u+4=0 gives u = 1 or 4, so x = ±1 or ±2.

It says "A system needs as many equations as unknowns (2 or 3), and a single equation can have only one unknown."

Either one equation has two unknowns (x+y=5 alone), or the number of equations differs from the number of unknowns, or you typed more than 3 equations. Add the missing equation after a semicolon, as in x+y=5; x-y=1, or remove the extra unknown. If you see "Systems must be linear: only terms like 3x or -2y, with no x² or xy.", the system has a squared or multiplied term, which this tool does not solve.

In the coefficient boxes it says "Cannot read “1e3” in a. Use a number such as 2, -0.5, 0,5, 1/2 or √2."

One of the boxes under "Or enter the coefficients" holds something that is not a number. The message names the box, and shows ∅ when the box is empty. Letters and scientific notation such as 1e3 are not accepted: write 1000 or 10^3. Fractions (1/2), decimal commas (0,5), roots (√2), powers (2^3), pi and brackets such as (1+2)/3 are fine. A box with a problem gets a rose-colored border.

On the Quadratic tab it says "For a quadratic, the coefficient a must not be 0."

With a = 0 the equation is not a quadratic any more. Put the same numbers into the "Linear" tab (b and c become a and b), or type the equation in "Type an equation", where the tool picks the type from the highest power it finds.

How it compares

Working a quadratic by hand with the formula gives the same roots, and the page follows the same steps, so it is useful for checking homework or seeing where a √ comes from. A graphing calculator or a computer algebra system goes further: it handles higher powers, trigonometric and exponential equations, inequalities, nonlinear systems and symbolic answers, none of which this page does. Its exact answers are limited to quadratics with whole-number or simple-fraction coefficients and cubics with a simple rational root; everything else is shown as decimals rounded to 4 places. When an equation is refused, the message links to the Graphing Calculator (for higher powers) or the Matrix Calculator (for systems).