# How do you simplify an expression?

Simplifies an expression, checking the result equals it at 20 points.

- Page: https://www.acalculator.org/math/simplify-calculator
- JSON spec: https://www.acalculator.org/math/simplify-calculator.json
- Version: 02ed42f3894e

## Default answer

Example with the default inputs (Expression (x^2 - 9)/(x^2 + 4x + 3)): (x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| expr | Expression | The expression, in one letter. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| simplified | Simplified | An equal, simpler expression. |

## Method

A computer algebra system rewrites the expression; the result shows only when it equals the expression at 20 points.

## Assumptions

- The result equals the expression wherever both are defined.
- An answer that fails its check is not shown.

## Worked examples

1. expr = (x^2 - 9)/(x^2 + 4x + 3) gives simplified = (x - 3)/(x + 1). Source: OpenStax College Algebra 2e, 1.6, Ex. 1. https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions.
2. expr = 6/(x^2 + 4x + 4) - 2/(x^2 - 4) gives simplified = 4 (x - 4)/((x - 2) (x + 2)^2).
3. expr = sin(x)^2 + cos(x)^2 gives simplified = 1.

## FAQ

### What does simplify mean?

Rewrite an expression in a shorter form that has the same value: collect like terms (3x + 2 − x + 5 = 2x + 7), cancel factors shared by a numerator and a denominator, combine fractions over a common denominator, and apply identities such as sin²x + cos²x = 1.

### How do you simplify a rational expression by hand?

Factor the numerator and the denominator completely, then cancel the factors they share. (x² − 9)/(x² + 4x + 3) = (x − 3)(x + 3)/((x + 1)(x + 3)) = (x − 3)/(x + 1). Note the values that made the original denominator 0 (here −1 and −3); they stay excluded.

### Is the simplified form equal to my expression everywhere?

Everywhere both are defined. Cancelling a factor can fill a hole: (x² − 9)/(x² + 4x + 3) has no value at x = −3, while (x − 3)/(x + 1) gives 3 there. Likewise e^(ln x) is only defined for x > 0, but simplifies to x.

### Why is my answer in a different form than the book?

There is no single simplest form. 6(x + 1) and 6x + 6 are both simple; the page shows the form the algebra finds. If you want a product, try the factoring calculator; for a sum of simple fractions, the partial fraction calculator.

### Can it simplify numbers too?

Yes: sqrt(8) gives 2 sqrt(2), 1/3 + 1/6 gives 1/2, and cos(pi/3) gives 1/2. The math calculator also evaluates a number expression to a decimal.

### What does "No verified answer" mean?

The result is compared with your expression at 20 points. If they differ anywhere, if the algebra returns a rounded number in place of an exact one, or if the work takes over 3 seconds, the page says "No verified answer" instead of showing a form it could not check.

## Sources

- OpenStax, College Algebra 2e, section 1.6 Rational Expressions: https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions
- OpenStax, Precalculus 2e, section 7.1 Solving Trigonometric Equations with Identities: https://openstax.org/books/precalculus-2e/pages/7-1-solving-trigonometric-equations-with-identities