Algebra Calculator

This algebra solver takes one equation in one unknown, finds every value that makes it true and shows the algebra a teacher expects to see. Linear and quadratic equations are solved with exact arithmetic, so x/3 + 2 = x/2 − 1 returns x = 18 and x² = 4x + 1 returns x = 2 ± √5 instead of rounded decimals. Equations without an algebraic formula, such as 2^x = 10, are solved numerically, and every such answer carries the ≈ sign.

Sister pages go deeper on single equation types. The Linear Equation Calculator focuses on ax + b = cx + d, the Quadratic Formula Calculator adds the vertex of the parabola, the Polynomial Calculator adds, subtracts and multiplies polynomials, the Order of Operations Calculator evaluates long numeric expressions, and the System of Equations Calculator solves two or three equations together. Use this equation calculator when you have a single equation of any type and want the answer, the steps and a check in one place.

How the solver reads and classifies an equation

Your input is first parsed into an expression tree, and the Read as line shows the result in standard notation so you can catch a typo or an unintended reading before trusting the answer. The solver then subtracts the right side from the left and looks at what is left, because that decides the method:

• Polynomial (only whole-number powers of x): expanded with exact fractions. Degree 1 is linear, degree 2 is quadratic, and degree 3 and up goes through the rational root test.
• Rational (x in a denominator): every denominator becomes a restriction, both sides are multiplied by the least common denominator, and the resulting polynomial equation is solved and checked against the restrictions.
• Anything else (x in an exponent, under a root, or inside abs, log or sin): solved numerically by scanning for sign changes.
• No x left after simplifying: an identity, true for all real numbers, or a contradiction with no solution.

The formulas behind every step

Each method rests on a short list of facts. The first line is the balance rule that every algebra step uses; the others are applied once an equation is in standard form:

The quadratic formula and its discriminant are the standard results listed in the NIST Digital Library of Mathematical Functions, which also states that a polynomial of degree n has exactly n roots when complex and repeated roots are counted. That is why the calculator always reports how many real and complex solutions it found.

Worked example 1: a linear equation with parentheses

Solve 3(x − 4) + 2 = 5x − 6. Distributing the 3 turns the left side into 3x − 12 + 2, and combining the constants leaves 3x − 10 = 5x − 6. There are x-terms on both sides and the right side has the larger coefficient, so the solver subtracts 3x from both sides to keep the coefficient positive:

Both sides equal −16, so x = −2 is the only solution. Gathering x on the right is a choice, not a rule: subtracting 5x first gives −2x − 10 = −6 and the same answer, with one more negative sign to manage.

Worked example 2: a quadratic that is not in standard form

Solve (x − 1)² = 3x + 7. Squaring the binomial gives x² − 2x + 1 = 3x + 7, and subtracting 3x + 7 from both sides puts the equation in standard form, x² − 5x − 6 = 0, with a = 1, b = −5 and c = −6. The discriminant is D = (−5)² − 4(1)(−6) = 25 + 24 = 49. Because 49 is a perfect square, both roots are rational:

The factored form (x − 6)(x + 1) = 0 confirms the pair. Substituting x = 6 gives (6 − 1)² = 25 on the left and 3·6 + 7 = 25 on the right; substituting x = −1 gives 4 on both sides. If the discriminant were not a perfect square, the calculator would simplify the radical instead: x² = 4x + 1 leads to D = 20, √20 = 2√5 and x = 2 ± √5, about 4.236068 and −0.236068.

Worked examples 3 and 4: a rational equation and an exponential one

Solve 1/x + 1/(x + 1) = 1/2. Before any algebra the solver records the restrictions x ≠ 0 and x ≠ −1, because either value would put a zero in a denominator. Writing the left side as one fraction gives (2x + 1)/(x² + x) = 1/2, and multiplying both sides by the common denominator 2x(x + 1) clears every fraction: 4x + 2 = x² + x. Rearranged, that is x² − 3x − 2 = 0 with D = 9 + 8 = 17, so x = (3 ± √17)/2, about 3.561553 and −0.561553. Neither value breaks a restriction, and either one makes the left side equal 0.5.

Now solve 2^x = 10. With x in the exponent the equation is not a polynomial, so the solver switches to its numeric method. It scans f(x) = 2^x − 10, finds a sign change between x = 3.3 (where 2^x ≈ 9.849) and x = 3.35 (where 2^x ≈ 10.196), and bisects that interval until the value is fixed to full precision: x ≈ 3.321928. By hand you would take logarithms of both sides, x = log 10 ÷ log 2 = 3.3219280949, which matches the numeric result in every displayed digit.

16 equations solved: type, answer and method

Every row below was produced by the calculator on this page at its default setting of six decimal places. Exact rows show integers, fractions, radicals or complex numbers; numeric rows show decimals.

Two rows deserve a closer look. The (x² − 1)/(x − 1) = 2 row reduces to x = 1, which the restriction check throws out, so an equation that seems to produce an answer correctly ends with no solution. The √(x + 3) = x − 3 row would give two candidates, x = 1 and x = 6, if you squared both sides by hand; only x = 6 satisfies the original equation, because √4 = 2 is not equal to 1 − 3 = −2.

How to use the calculator and type your equation

Pick a mode and type into the box; the result updates as you type. Solve needs exactly one equals sign, and if you leave it out the expression is set equal to 0. Simplify expands, combines like terms and cancels common factors in a fraction, so 2x + 3(x − 4) − x² + 5x² becomes 4x² + 5x − 12 and (x² − 9)/(x − 3) becomes x + 3 with the note x ≠ 3. Evaluate substitutes a value: 3x² − 2x + 1 at x = 4 is 48 − 8 + 1 = 41. Type numbers without thousands separators (1000, not 1,000), because the comma also separates the two inputs of log(x, b); the calculator points this out instead of guessing. The table shows how common inputs are read, which is where most surprises come from:

Who this calculator is for

Students in pre-algebra, Algebra 1 and Algebra 2 use it to check homework and to compare a clean set of steps with their own. Parents and tutors use the step list to find where a solution went wrong, since a single sign error shows up as a mismatch in the substitution check. Test-prep students working toward the SAT, ACT, GED or a placement exam use it to drill equation types quickly. Anyone who needs a math equation solver for a practical problem, such as the month when two price plans cost the same or the rate in a growth formula, can type the equation directly. People who search for a variable calculator or a solve for x calculator usually want exactly this: one unknown, solved and checked.

Key terms

• Variable (unknown): the letter you solve for; any single letter except i works.
• Coefficient: the number multiplying a power of the variable, such as 5 in 5x².
• Like terms: terms with the same power of the variable, which can be added together.
• Degree: the highest power of the variable after simplifying; it decides which method applies.
• Discriminant: D = b² − 4ac for a quadratic; positive means two real roots, zero means one repeated root, negative means a complex pair.
• Root or solution: a value of the variable that makes both sides equal.
• Multiplicity: how often a root repeats; in (x − 1)²(x + 2) = 0, x = 1 is a double root.
• Identity and contradiction: an equation that is true for every value of the variable, and one that is true for none.
• Restriction and extraneous solution: a value excluded because it makes a denominator zero, and a candidate that fails the original equation.

Exact or numeric: what changes the answer

The same keystrokes can produce different kinds of answer, and four things decide which:

• The type of equation. Polynomials of degree one and two always get exact answers. From degree three up, rational roots and factors that reduce to quadratics are exact; the rest are decimals.
• Where the parentheses go. 1/(2x) = 4 gives x = 1/8, while 1/2x = 4 gives x = 8, because the second is read as one half times x.
• Decimals in the input. 0.5x + 1.25 = 2 is still solved exactly, because the decimals are converted to fractions, and the answer is shown as 1.5 with 3/2 beside it.
• Restrictions. In a rational equation, a value that makes a denominator zero is removed even when the algebra produces it.

Tips for solving equations by hand

• Clear fractions first: multiplying x/3 + 2 = x/2 − 1 by 6 gives 2x + 12 = 3x − 6 with no fractions left.
• Gather the x-terms on the side where the coefficient is larger, so it stays positive.
• Move everything to one side before factoring a quadratic, and never divide both sides by x: x² = 3x has two solutions, 0 and 3.
• Simplify radicals and keep exact forms until the very last step.
• Check every answer in the original equation, not a rearranged version, so extraneous solutions show up.

Limitations

The solver handles one equation in one unknown. It does not solve systems, inequalities or a formula for one letter in terms of others, so substitute numbers for the other letters first. Numeric answers come from a search between −1,000,000 and 1,000,000, so a solution outside that range, or two solutions closer together than the 0.05 grid near zero, can be missed. Periodic equations such as sin(x) = 0.5 have infinitely many solutions, and the calculator lists the 12 closest to 0. Fractional powers are principal roots, so x^(1/3) is undefined for negative x; type cbrt(x) for a real cube root. Polynomials are limited to degree 60, and at very high degrees some complex roots cannot be computed accurately enough to list.

Related calculators

• For ax + b = cx + d with fraction answers, use the Linear Equation Calculator.
• For the discriminant, vertex and roots of one quadratic, use the Quadratic Formula Calculator.
• To add, subtract or multiply polynomials, use the Polynomial Calculator.
• To evaluate a long numeric expression in PEMDAS order, use the Order of Operations Calculator.
• For two or three equations with two or three unknowns, use the System of Equations Calculator.
• To factor a trinomial or a difference of squares, use the Factoring Calculator.
• For linear inequalities and interval notation, use the Inequality Calculator.
• To add, multiply or divide complex roots, use the Complex Number Calculator.

Sources

Algebra needs no outside data: every step is a rule of arithmetic applied to your own equation. The formulas and the software behavior described on this page can be checked against these primary references:

• NIST Digital Library of Mathematical Functions, §1.11 Zeros of Polynomials: the quadratic formula with D = b² − 4ac (equation 1.11.11), the statement that a polynomial of degree n has exactly n zeros counting multiplicity, Horner's scheme for synthetic division, and the cubic and quartic formulas.
• mathjs documentation, expression syntax: implicit multiplication takes precedence over division except in the number/number pattern, which is why 1/2x is read as (1/2)x.
• mathjs documentation, polynomialRoot: closed-form numeric roots of linear, quadratic and cubic polynomials, used for cubic factors that have no rational root.
• Worked examples and tables: every value on this page was produced by the calculator above and confirmed by substitution.