How do you simplify an expression?
Type an expression such as (x^2 - 9)/(x^2 + 4x + 3) to get a simpler form. The page shows it only after checking it equals your expression at 20 points.
- Simplified
- (x - 3)/(x + 1)
(x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).
Simplified: (x - 3)/(x + 1). (x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).
How to calculate
Simplifies an expression, checking the result equals it at 20 points.
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).
Method: A computer algebra system rewrites the expression; the result shows only when it equals the expression at 20 points.
- The result equals the expression wherever both are defined.
- An answer that fails its check is not shown.
Worked examples
Each example is checked against the calculator on every build.
- Expression (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
- Expression 6/(x^2 + 4x + 4) - 2/(x^2 - 4) gives Simplified 4 (x - 4)/((x - 2) (x + 2)^2).
- Expression sin(x)^2 + cos(x)^2 gives Simplified 1.
How it works
The simplify calculator rewrites your expression in a simpler form with a computer algebra system (nerdamer, open source): it collects like terms, cancels common factors, combines fractions, and applies identities such as sin²x + cos²x = 1. The algebra runs after you start typing, in the background.
Every answer is checked before it is shown. The result must equal your expression at 20 points where both are real numbers (fixed pseudo-random numbers, mostly between −4 and 4, some up to ±10), to 1 part in 10⁹. A result that fails is never shown; the page says "No verified answer" instead.
What you can type
- One variable. Use one lowercase letter: x, t, y, and so on. Two different letters give no answer. The letter e is Euler’s number, and pi (or π) is π. An expression with no letter is simplified as a number.
- Numbers can have decimals (2.5) and powers of ten (1e-3).
- Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, 3(x + 1), x sin(x).
- Functions: sqrt, cbrt, ln (and log, the same natural logarithm), log10, exp, abs (or |x|), sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin, arccos, arctan also work), sinh, cosh, tanh and their inverses. Angles are in radians. sin x means sin(x); sin x^2 means sin(x²).
How answers are written
- The result is in the syntax you type: ^ for powers, sqrt(u), |u| for absolute value, π for pi, ln for the natural logarithm, a/b for a fraction, a space or a number before a letter for multiplication.
- Terms go by power of the letter, highest first; the factors of a product go lowest degree first:
4 (x - 4)/((x - 2) (x + 2)^2). - √(x²) is |x|, not x. The result is a real number wherever your expression is: when the algebra narrows the domain (it rewrites ln(x²) as 2 ln(x), which has no value for x < 0), each ln(u) becomes ln|u| (2 ln|x|), and the page checks the result again; if that does not restore the domain, there is no answer.
- An answer holding a number the algebra rounded (a fraction whose denominator, after removing factors 2 and 5, is over 1,000,000, or whose numerator times that denominator is over 10¹², or a decimal with more than 12 significant digits, or a whole number past 2⁵³) is not shown.
Assumptions
- The simplified form equals the expression wherever both are defined; it may be defined at points where the expression is not.
- Only real values count. Angles are in radians; ln and log are the natural logarithm.
- An answer that fails its check or takes over 3 seconds is not shown ("No verified answer").
Worked examples by hand
(x² − 9)/(x² + 4x + 3) (OpenStax College Algebra 2e, section 1.6, Example 1). Factor: x² − 9 = (x − 3)(x + 3) and x² + 4x + 3 = (x + 1)(x + 3). Cancel x + 3: (x − 3)/(x + 1), for x ≠ −3.
6/(x² + 4x + 4) − 2/(x² − 4) (OpenStax College Algebra 2e, section 1.6, Example 5). Factor the denominators: (x + 2)² and (x + 2)(x − 2). The common denominator is (x + 2)²(x − 2). Then 6(x − 2) − 2(x + 2) = 6x − 12 − 2x − 4 = 4x − 16 = 4(x − 4), so the result is 4(x − 4)/((x − 2)(x + 2)²).
sin²(x) + cos²(x) (the Pythagorean identity, OpenStax Precalculus 2e, section 7.1). For every angle x, a point (cos x, sin x) lies on the unit circle, so cos²x + sin²x = 1.
Other questions people ask
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.