Algebra Calculator with Exact Results
Simplify polynomial expressions, expand parentheses, combine like terms, evaluate an expression at x, or solve a one-variable linear equation. Exact fractions stay exact through every step.
Last Updated: July 26, 2026
Online Algebra Calculator
Choose a task, enter an expression or equation using x, then review the simplified form, exact result, coefficient table and calculation steps.
Your simplified expression or equation solution will appear here.
Calculation steps
- Enter an algebraic expression or linear equation.
Coefficient table
| Power of x | Exact coefficient | Term |
|---|---|---|
| Calculate an expression to see its polynomial coefficients. | ||
Recent calculations
- Your last six results will appear here.
How to Use This Algebra Calculator
- Choose Simplify or evaluate an expression when you want to expand parentheses, combine like terms or substitute a value for x.
- Choose Solve a linear equation when the problem contains one equals sign and the highest remaining power of x is 1.
- Type the expression or equation. You may write 3x, 3*x, 2(x + 1), fractions such as 1/2 and powers such as x^2.
- Add an optional x value in simplify mode when you need a numerical evaluation.
- Select Calculate Algebra. Review the standard form, exact answer, decimal display, coefficient table and steps.
Use x as the variable. The tool accepts integers, finite decimals, fractions, scientific notation, parentheses and nonnegative whole-number powers. It interprets multiplication written next to a variable or parenthesis, so 4x and 4*x mean the same thing. Do not enter commas inside numbers. Write 2500 instead of 2,500.
What This Algebra Calculator Does
This calculator works with one-variable polynomial expressions. It converts each supported input into exact coefficients, expands multiplication, applies powers, combines terms with the same exponent and orders the final terms from highest power to constant. If you provide x, it evaluates the simplified polynomial with that exact value.
The linear equation mode simplifies both sides separately. It then moves every term to the left, producing a standard equation in the form ax + b = 0. When a is not zero, the unique solution is x = -b/a. The tool substitutes the answer into both original sides for an exact verification.
The page intentionally stays narrower than the dedicated Equation, Quadratic Equation, Factoring and Graphing calculators listed in the Math & Statistics plan. It does not claim to solve every symbolic problem. This clear boundary gives you a dependable result for the supported operations instead of a guessed answer for unsupported notation.
Supported Algebra Input
| Input type | Example | What the calculator does |
|---|---|---|
| Like terms | 3x + 2x - 4 | Combines the x terms to return 5x - 4. |
| Implicit multiplication | 2(x + 3) | Treats the adjacent terms as multiplication and expands them. |
| Polynomial product | (x + 1)(x - 1) | Multiplies the factors and returns x^2 - 1. |
| Exact fraction | (1/2)x + (3/4)x | Combines exact coefficients to return (5/4)x. |
| Substitution | 2x + 5 at x = 3 | Simplifies first, then returns the exact value 11. |
| Linear equation | 3x + 2 = 14 | Rearranges to 3x - 12 = 0 and solves x = 4. |
Combining Like Terms
Like terms have the same variable raised to the same power. The terms 7x and -2x are like terms because each contains x to the first power. Their coefficients combine: 7 + (-2) = 5, so 7x - 2x becomes 5x. A constant such as 4 does not combine with an x term.
The same rule applies to higher powers. The terms 3x^2 and (1/2)x^2 combine to (7/2)x^2. However, x^2 and x are not like terms. Their exponents differ, so they remain separate in standard form.
= (4 - 1)x^2 + (3 + 5)x - 7
= 3x^2 + 8x - 7
Using the Distributive Property
The distributive property multiplies every term inside a group by the factor outside it. For a(b + c), the equivalent expanded form is ab + ac. Missing one inside term is a common source of wrong answers.
= 2x + 6 + 4x - 5
= 6x + 1
Products of two polynomial groups use the same idea repeatedly. Multiplying (x + 1)(x - 1) gives x^2 - x + x - 1. The middle terms cancel, leaving x^2 - 1. The coefficient table helps you see this cancellation because the coefficient of x becomes zero.
Order of Operations in Algebra
Grouping and exponent rules still control algebraic work. Simplify parentheses first, apply powers, perform multiplication and division, then add or subtract. The calculator follows that precedence. For example, -x^2 means the negative of x squared, while (-x)^2 squares the complete negative expression.
Division is supported only when the divisor simplifies to a nonzero constant. This means (x + 2)/2 is supported, but 1/x is not. Dividing by a variable creates a rational expression with a domain restriction, which falls outside this polynomial tool.
Worked Example: Simplify and Evaluate
Simplify 2(x + 3) + 4x - 5, then evaluate it at x = 3. Expand the parenthesis first. Combine x terms and constants next. Substitute only after the expression is simplified.
= 2x + 6 + 4x - 5
= 6x + 1
At x = 3:
6(3) + 1 = 18 + 1 = 19
The simplified expression is 6x + 1. Its degree is 1, leading coefficient is 6 and constant term is 1. Evaluation produces the exact value 19. Keeping simplification and substitution as separate stages makes the work easier to check.
How the Linear Equation Solver Works
A linear equation has a variable only to the first power after both sides are simplified. The calculator expands and combines each side, subtracts the right side from the left, and writes the result as ax + b = 0.
ax = -b
x = -b/a, where a is not 0
For 3(x - 2) + 4 = 2x + 9, the left side simplifies to 3x - 2. Moving the right side left gives x - 11 = 0. Therefore x = 11. Substitution confirms that both original sides equal 31.
One Solution, No Solution or Infinitely Many
A simplified linear equation does not always produce one number. If the x coefficients cancel and different constants remain, the equation is a contradiction with no solution. For example, 2x + 1 = 2x + 3 becomes -2 = 0, which is false for every x.
If both variable and constant terms cancel, the equation is an identity. The equation 2(x + 1) = 2x + 2 becomes 0 = 0, so every real x satisfies it. The tool labels these outcomes directly instead of dividing by zero or returning a misleading number.
Exact Fractions, Decimals and Rounding
Finite decimals are converted to fractions based on their written place value. For example, 0.125 becomes 1/8. A coefficient written as 1/3 stays 1/3. Polynomial operations use reduced fractions backed by integers, so combining 1/2 and 3/4 returns 5/4 without binary floating-point drift.
The decimal-place selector affects only the optional decimal display. If an exact value repeats, the displayed decimal includes an approximation mark. Use the exact fraction for later algebraic work. Round only when the final task requires a decimal with a stated precision.
Polynomial Degree and Coefficients
The degree is the highest exponent with a nonzero coefficient. In 5x^3 - 2x + 7, the degree is 3 and the leading coefficient is 5. The constant term is 7. For a nonzero constant, the degree is 0. The zero polynomial has no single conventional degree or defined leading coefficient, so the calculator labels those fields separately.
The coefficient table lists every nonzero term. Reading the table makes it easier to compare expressions, verify standard form and notice canceled powers. A missing row for x^2 means its combined coefficient is zero.
Limits and Error Messages
This tool uses one variable named x and supports polynomial degree up to 12 after expansion. An entered exponent must be a nonnegative whole number no greater than 12. Large products that would exceed the limit are stopped before they create an unwieldy result. The indeterminate form 0^0 is rejected, while x^0 simplifies to 1 when the base is not already the zero polynomial.
Functions such as sin, log and sqrt belong in the Scientific Calculator. Variable denominators, negative powers and radical variables are rational or non-polynomial expressions. Quadratic root solving, general equation solving, factoring and graphing have separate planned tools. An error message explains these scope limits without changing your input.
Common Algebra Mistakes
- Combining terms with different powers, such as x and x^2.
- Distributing a factor to only the first term inside parentheses.
- Dropping a negative sign before a grouped expression.
- Substituting a value before simplifying, then losing parentheses around a negative value.
- Dividing both sides by a coefficient that has simplified to zero.
- Rounding a fraction early and carrying the rounded value into later steps.
- Entering a quadratic equation in the linear mode and expecting two roots.
Method and Review Basis
The expression rules follow the order of operations and algebra essentials described by OpenStax College Algebra 2e. The polynomial degree, coefficient and operation definitions follow OpenStax Polynomials. The equation steps use the general linear strategy in OpenStax Intermediate Algebra 2e.
Every supported calculation is reduced to exact polynomial coefficients. Equation answers are checked by substitution into both original sides. Use the displayed steps to review the algebra, and follow the notation or method required by your class, textbook or professional workflow.
Related Calculators
Frequently Asked Questions
What does this algebra calculator simplify?
It simplifies supported one-variable polynomial expressions by expanding parentheses, applying nonnegative whole-number powers and combining like terms.
Can I type 3x instead of 3*x?
Yes. The calculator recognizes multiplication next to x or parentheses. Both 3x and 3*x mean three times x.
Does it solve equations with x on both sides?
Yes, when simplifying both sides produces a linear equation. It moves all terms to one side, solves the exact coefficient equation and checks the answer.
Does it solve quadratic equations?
No. It may expand and simplify a polynomial containing x^2, but the linear equation mode does not calculate quadratic roots.
Can I evaluate an expression for a value of x?
Yes. Choose simplify mode, enter the expression and add an optional x value. The tool substitutes that value after simplifying.
Are fraction coefficients supported?
Yes. Enter fractions such as 1/2 or -3/4. The calculator keeps reduced exact fractions through expansion, simplification and equation solving.
Why is division by x rejected?
Division by x creates a rational expression and requires a domain restriction. This calculator supports division only by a nonzero constant polynomial.
What is a like term?
Like terms contain the same variable raised to the same power. Their coefficients may be added or subtracted while the variable part stays unchanged.
What does no solution mean?
It means the variable terms cancel but unequal constants remain, creating a false statement such as 1 = 3.
What does infinitely many solutions mean?
It means both sides simplify to the same expression. The equation becomes an identity that is true for every real value of x.
Why does the decimal answer show an approximation mark?
The exact fraction does not terminate within the selected decimal places. The fraction remains exact while the decimal is rounded for display.
Is my algebra input sent to a server?
No input is needed by a server to calculate the answer. The JavaScript calculation runs locally in your browser.
Disclaimer: This calculator provides general mathematical results for learning and checking work. Verify notation, domain restrictions, assumptions and required rounding before using an answer for academic, scientific, financial, engineering or other important decisions.