An algebraic expression is a math phrase built from numbers, variables, and operations — 3x plus 5, for example — and the one thing it never has is an equals sign. That missing equals sign is the whole distinction: an equation states that two things are equal and asks you to solve for the unknown, while an expression is just a quantity waiting to be simplified, expanded, factored, or evaluated. This page teaches all four operations, because students search for one of them and then discover they needed another.

The vocabulary.

Every expression is made of terms, and every term has a coefficient and a variable part. In 3x squared plus 5x minus 7, the terms are 3x squared, 5x, and negative 7. The coefficient is the number in front: 3, 5, and negative 7 is a constant term, a term with no variable at all. The exponent matters as much as the letter: x squared and x are different variable parts, which is why they are not interchangeable. Two terms are called like terms when their variable parts match exactly, including exponents: 3x squared and negative 2x squared are like terms, while 3x squared and 5x are not, no matter how similar they look. Almost every simplification mistake starts here, with terms combined that were never like terms to begin with.

Combining like terms.

Simplifying usually means collecting the like terms into fewer, cleaner ones. Take 3x squared plus 5x minus 2x squared plus 7x. Group the x-squared terms: 3x squared minus 2x squared is x squared. Group the x terms: 5x plus 7x is 12x. The simplified form is x squared plus 12x. A simpler case: 2x plus 3x minus 5 plus 10 becomes 5x plus 5, because 2x plus 3x is 5x and negative 5 plus 10 is 5. Notice what does not combine: 3x plus 4y stays 3x plus 4y forever, because x and y are different variables, and no amount of wishing merges them. Neither does 3x plus 4: a variable term and a constant are never like terms. When students ask why their simplified answer still has two terms, this is usually why — the terms left over are genuinely different kinds of things.

The distributive property.

Parentheses are instructions to multiply, and the distributive property is how you carry them out. Multiply each term inside by the number outside: 2 times x plus 3 is 2x plus 6. Combine that with the rest of an expression and you get the classic worked example: 2 times x plus 3, plus 4x, distributes to 2x plus 6 plus 4x, then combines to 6x plus 6. Two binomials multiply by distributing every term of the first across every term of the second: x plus 2 times x plus 3 gives x squared plus 3x plus 2x plus 6, which combines to x squared plus 5x plus 6. The most useful special case is the difference of squares: x plus 1 times x minus 1 gives x squared minus x plus x minus 1, and the middle terms cancel, leaving x squared minus 1. Recognizing that pattern saves real time, because difference-of-squares pairs appear constantly in factoring, in fractions, and in rationalizing denominators.

Factoring: simplification in reverse.

Factoring undoes expansion: it rewrites a sum as a product. Always look for a greatest common factor first. In 14y plus 21a, both terms share 7, so the factored form is 7 times 2y plus 3a — and expanding it back gives the original, which is the check you should always run. For quadratics, look for two numbers that multiply to the constant and add to the x-coefficient: x squared plus 5x plus 6 factors as x plus 2 times x plus 3, because 2 times 3 is 6 and 2 plus 3 is 5. Difference of squares factors instantly: x squared minus 9 is x minus 3 times x plus 3. With four terms, group them: 2x cubed plus 6x squared minus 8x minus 24 groups as 2x squared times x plus 3, minus 8 times x plus 3, which shares the factor x plus 3 and becomes 2 times x minus 2 times x plus 2 times x plus 3. Each factoring method is just expansion read backwards, and verifying by expanding is the habit that separates reliable answers from hopeful ones.

Evaluating: putting numbers in.

Evaluating means replacing each variable with a number and doing the arithmetic. For 2x plus 4 when x is 3: substitute to get 2 times 3 plus 4, which is 6 plus 4, which is 10. For 3x squared minus 2x plus 5 when x is 4: 3 times 16 minus 8 plus 5 is 48 minus 8 plus 5, which is 45. One notation trap deserves emphasis: 2x squared means 2 times x squared, not 2x squared as a group. At x equals 3, 2x squared is 2 times 9, which is 18 — while 2x all squared would be 36. The exponent binds to the variable alone unless parentheses say otherwise. Negative substitutions need parentheses too: evaluating x squared at x equals negative 3 gives negative 3 squared, which is 9, but writing it without parentheses as negative 3 squared invites reading it as negative 9, which is wrong. Two-variable expressions work the same way, one substitution at a time: for 2a plus 3b with a equals 4 and b equals 5, substitute to get 2 times 4 plus 3 times 5, which is 8 plus 15, which is 23. Keep each product separate until the final addition, and sign errors have nowhere to hide.

Fractions inside expressions.

Expressions with fractions simplify by the same rules, with one extra caution about division by zero. The fraction 12x over 18 reduces like any fraction: divide numerator and denominator by 6 to get 2x over 3. A rational expression like x squared minus 4, over x minus 2, factors on top as x minus 2 times x plus 2, so the x minus 2 factors cancel and leave x plus 2 — but only with the condition that x is not 2, because the original fraction is undefined there. Cancelled factors do not erase restrictions: the simplified form and the original agree everywhere except at the cancelled value, where the original never existed. Teachers test exactly this, so write the restriction down every time you cancel.

How to use the calculator above.

The calculator on this page evaluates the product of two values, which is the operation at the center of evaluating any expression once you substitute numbers. Evaluating 3x at x equals 7 comes down to 3 times 7: enter 3 as Variable A and 7 as Variable B, and the Result is 21. The page's own example uses 10 and 5, giving 50. Use it for the multiplication step inside a longer evaluation: for 3x plus 5 at x equals 4, the product step is 3 times 4, which is 12, and adding 5 gives 17. For a two-variable product like 6ab with a equals 3 and b equals 8, the product step is 6 times 3 times 8 — do it as two calculator steps, 6 times 3 is 18, then 18 times 8 is 144. The calculator handles the arithmetic; the algebra — deciding which numbers to multiply — is the skill this page teaches.

Mistakes that break simplifications.

The greatest-hits list, in one place. First, the square-of-a-sum error: x plus y squared is x squared plus 2xy plus y squared, not x squared plus y squared — the middle term is real and it is usually the whole point of the problem. Second, sign errors when distributing a negative: negating the bracket 3x minus 2 gives negative 3x plus 2, because the negative flips both signs inside. Third, dropping terms when expanding: 4 times a plus 5, minus 2 times 3a plus 6, expands to 4a plus 20 minus 6a minus 12, which is negative 2a plus 8 — students who forget to distribute the negative 2 to the 6 lose twelve points of value. Fourth, combining unlike terms: 3x plus 4y is already simplified, and 5x plus 7 is already simplified. Fifth, forgetting that the exponent applies to the variable only: 3x squared at x equals 2 is 12, not 36. Check every simplification by substituting a number — x equals 2 is usually enough — into both the original and your answer. If they disagree, the simplification is wrong, and the disagreement tells you exactly where to look.

Why bother simplifying.

Simplification is not tidiness for its own sake; it is what makes the next step possible. An equation cannot be solved until its sides are simplified. A graph cannot be sketched until the expression's structure is visible — factored form shows the zeros, expanded form shows the y-intercept. A fraction cannot be reduced until numerator and denominator are factored. And in applied work, a simplified expression is a cheaper computation: evaluating x squared plus 12x at a hundred data points beats evaluating the unsimplified five-term version every time. Learn to simplify fluently and every later algebra topic — equations, functions, quadratics, rational expressions — gets easier, because all of them assume you already can.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Algebraic Expression Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.