Math Solver

How the solver works

The tool reads what you type as a math expression, using the same order of operations you learned in school: parentheses first, then exponents, then multiplication and division, then addition and subtraction. If your input contains an equals sign, it treats the two sides as an equation to solve rather than a single expression to evaluate, and figures out on its own whether that equation is linear (the variable appears on its own, like 2x + 3 = 7) or quadratic (the variable is squared somewhere, like x^2 - 5x + 6 = 0).

For plain arithmetic, the tool narrates each reduction it performs, working from the innermost parentheses outward, exactly the order a calculator (or a careful student) would follow by hand. For equations, it moves every term to one side, expands and combines like terms, and then applies the appropriate method (isolating the variable for a linear equation, or the quadratic formula for a quadratic one) with each intermediate calculation shown.

What you can type

The input box accepts a single line of text. A few things to know about the notation it expects:

• Use * for multiplication and / for division: 3 * 4, not 3 x 4.
• Use ^ for powers: x^2 means x squared, 2^3 means 2 cubed.
• Parentheses work exactly like on paper: 2 * (3 + 4).
• Function names are supported for simple cases, such as sqrt(16).
• For an equation, include exactly one equals sign: 2x + 3 = 7. For a plain expression, leave the equals sign out entirely.
• Only one variable letter per equation is supported right now (commonly x, but any single letter works).

Worked example: a linear equation

Say you type 2x + 3 = 7. The solver first moves everything to one side, turning the equation into 2x + 3 - (7) = 0, then expands and combines the constant terms to get 2x - 4 = 0. From there it isolates the variable: 2x = 4, then divides both sides by the coefficient of x to get x = 4 / 2, which simplifies to the final answer.

See the examples page for more worked problems, including cases where the variable appears inside parentheses on both sides.

Worked example: a quadratic equation

For x^2 - 5x + 6 = 0, the solver recognizes the squared variable and switches to the quadratic method. After expanding, it writes the equation in standard form (a·x^2 + b·x + c = 0), here with a = 1, b = -5, c = 6. It then computes the discriminant, b^2 - 4ac, which works out to (-5)^2 - 4(1)(6) = 1. Because the discriminant is positive, there are two real solutions, found with the quadratic formula: x = (-b ± √discriminant) / (2a). Plugging in the numbers gives x = 3 or x = 2, which you can check by noticing that (x - 2)(x - 3) expands back to the original equation.

Not every quadratic has two solutions. When the discriminant works out to exactly zero, there is a single repeated solution. When it is negative, there are no real solutions at all, and the tool says so plainly instead of showing a broken result.

Worked example: order of operations

For a plain expression like 12 * (3 + 4) / 2 (no equals sign), the solver evaluates the parentheses first: 3 + 4 becomes 7, turning the expression into 12 * 7 / 2. Multiplication and division are then applied left to right: 12 * 7 becomes 84, and 84 / 2 gives the final answer, 42. Each of those intermediate expressions is shown as its own step, so you can see exactly where a different answer would come from a different order of operations, a common source of arithmetic mistakes.

Why order of operations trips people up

Most arithmetic mistakes are not really about not knowing how to multiply or add, they are about doing the steps in the wrong order. 2 + 3 * 4 is 14, not 20, because multiplication happens before addition even though addition is written first. Parentheses exist specifically to override that default order when a problem needs a different sequence: (2 + 3) * 4 really is 20, because the parentheses force the addition to happen first.

Two rules cover almost every case: work from the innermost parentheses outward, and within a single level, do multiplication and division before addition and subtraction, working left to right. Exponents and roots are evaluated before multiplication and division. The step-by-step output above exists specifically to make that sequence visible instead of leaving it as something you have to track in your head. See the order of operations reference for the full rule, including nested parentheses and negative signs.

Linear vs. quadratic: what the difference actually means

An equation is linear when the variable never gets multiplied by itself. Graphed, a linear equation traces a straight line, and it has exactly one solution (unless it turns out to be always true or never true, covered below). A quadratic equation has the variable squared somewhere, and its graph is a curve (a parabola), which is why it can have zero, one, or two solutions depending on where that curve crosses zero.

When an equation has no solution, or infinitely many

Not every equation resolves to a single clean answer. Two special cases come up often enough to be worth calling out by name. If the variable cancels out completely and what is left is a true statement (for example, x - x + 2 = 2 reduces to 0 = 0), the equation is true for every possible value of the variable, meaning it has infinitely many solutions rather than one specific answer. If instead the variable cancels out and what is left is a false statement (for example, 5 = 3, or an equation that reduces to that), the equation is a contradiction: no value of the variable can ever make it true, so it has no solution at all. The solver detects both cases automatically and explains which one applies instead of guessing at a numeric answer that does not exist.

What the tool does not solve yet

This is a focused tool, not a full computer algebra system, and it is worth being upfront about the current limits so you are not left guessing why an input did not work. Equations with the variable raised to the third power or higher (cubic equations and beyond) are outside its current scope, and it will say so rather than returning a wrong or incomplete answer. The same is true for equations with more than one variable (such as 2x + y = 10) and for equations that mix a variable with a square root, trigonometric function, or logarithm in a way that cannot be expanded into a plain polynomial. In every one of these cases the tool reports plainly that the input is outside what it currently supports, rather than silently producing a guess. Two other limits are worth naming directly: the tool only accepts typed text, so it cannot solve a problem from a photo, screenshot, or scanned image, and it does not yet handle calculus, geometry, or statistics problems, covering only arithmetic and the equation types described above.

A note on precision

Whenever a solution is a whole number, the tool shows it as one. When a solution is a fraction that does not reduce to a whole number, such as x = 1/3, it shows both the exact reduced fraction and a rounded decimal (≈ 0.333333), so you get the exact value for further calculation and a quick decimal reading at a glance. For arithmetic (not equations), results are rounded to six decimal places to avoid the long floating-point tails that raw computer arithmetic sometimes produces, such as displaying 0.30000000000000004 instead of the expected 0.3.

A closer look at the quadratic formula

The quadratic formula solves any equation of the form a·x^2 + b·x + c = 0, and it is worth understanding where the discriminant, the part under the square root, actually comes from. The formula itself, x = (-b ± √(b^2 - 4ac)) / (2a), is derived by completing the square on the general quadratic, a process that rewrites the left-hand side as a perfect square plus a leftover constant. The discriminant is that leftover piece, and its sign alone determines how many real solutions exist, without needing to finish the rest of the calculation first.

A positive discriminant means the perfect-square term can equal a positive number in two different ways (a positive or a negative square root), giving two distinct real solutions. A discriminant of exactly zero means the perfect-square term can only equal zero, giving one repeated solution, which is also the vertex of the parabola touching the x-axis at a single point. A negative discriminant means no real number squared can produce a negative result, so the equation has no real solutions, though it does have two complex (imaginary) ones if you allow those, which is why some textbooks describe this case as having 'complex conjugate roots' rather than simply no solution.

Checking an answer by hand

It is good practice to verify a solved equation by substituting the answer back into the original problem, and it only takes a moment. For the linear example above, x = 2 substituted back into 2x + 3 = 7 gives 2(2) + 3 = 4 + 3 = 7, which matches the right-hand side exactly, confirming the solution. For the quadratic example, both x = 3 and x = 2 should make x^2 - 5x + 6 equal zero: (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0, and (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0. Both check out.

This habit catches two kinds of mistakes at once: a typo in the original problem (the check will fail even if every subsequent step was done correctly), and an arithmetic slip partway through the solution. It is a useful habit to build even when using a tool that shows its work, since it turns 'trust the tool' into 'verify the tool', a small extra step that builds real confidence in the answer rather than just borrowing it.

Common mistakes this solver helps you avoid

A handful of small errors account for most wrong answers in algebra and arithmetic, and seeing every step makes each one easier to catch.

• Dropping a negative sign when moving a term across the equals sign: 2x + 3 = 7 becomes 2x = 7 - 3, not 2x = 7 + 3.
• Forgetting to distribute a number across parentheses: 2(x + 3) is 2x + 6, not 2x + 3.
• Mixing up the order of operations, treating 2 + 3 * 4 as (2 + 3) * 4 instead of 2 + (3 * 4).
• Applying the quadratic formula with the wrong sign for b, especially when the middle term is already negative, as in x^2 - 5x + 6.
• Stopping after finding one root of a quadratic equation and forgetting the second one entirely.
• Assuming an equation has no solution just because the first attempt at solving it did not simplify cleanly, rather than checking whether it is actually an identity or a contradiction.