What is the answer to a math problem?

How it works

Type a problem and choose a mode. A computer algebra system (nerdamer, open source) does the algebra after you start typing, in the background, and each answer is shown only after a numeric check against what you typed. An answer that fails is never shown; the page says "No verified answer" instead.

• Evaluate (numbers only). The Value is computed directly from what you typed, in 64-bit floating point, and shown to 10 significant figures; no algebra is needed. An expression with a letter, or with no real value (a division by 0, the square root or logarithm of a negative number), gets no answer.
• Simplify. A simpler expression, equal to yours at 20 points where both are real, to 1 part in 10⁹. It may be defined where yours is not: (x² − 9)/(x² + 4x + 3) simplifies to (x − 3)/(x + 1), which has a value at x = −3. Here the result can also lose part of your expression's domain, as the algebra writes it: ln(x²) may show as 2 ln(x), which has no value for x < 0. The simplify calculator restores the domain (2 ln|x|).
• Solve. Every real solution of an equation left = right (with no = sign, of expression = 0), by the rules of the equation solver: the algebra solves the equation with its fractions cleared; a root counts when left − right is exactly 0 there (an exact root) or changes sign across it; roots where a side is not real are excluded; and a scan of left − right from −10¹² to 10¹² adds any solution the algebra missed, as a decimal marked ≈. A near-touch the scan cannot confirm, or a solution too close to the edge of the domain to compute, gives "No verified answer"; solutions that fill an interval, or an identity, give "The solutions fill an interval." Equations with sin, cos, tan, sec, csc or cot of the variable get no answer.
• Derivative. Checked against a five-point central difference of f at 20 points, to 1 part in a million (the derivative calculator also expands the answer and gives its value at a point).
• Integral. An antiderivative + C whose numeric derivative equals f at 20 points, to 1 part in a million, with ln|u| where the algebra gives ln(u) and both check (see the integral calculator, which also does definite integrals).

For limits and Taylor polynomials, use the limit calculator and the Taylor series calculator; for factoring, the factoring calculator.

What you can type

• Numbers with decimals (2.5) and powers of ten (1e-3). Constants: pi (or π) and e.
• One variable for Simplify, Solve, Derivative and Integral: any one lowercase letter (x, t, y, …) except e; with no letter the variable is x. Two different letters give no answer.
• Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, 3(x + 1), x sin(x). An = sign only in Solve mode, once.
• 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).
• Only real numbers count. A fractional power with an odd denominator is the real root: (−8)^(1/3) is −2.

How answers are written

• Simplified, Derivative and Integral are written as the algebra gives them, 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. The terms may come in any order (3x^2/2 - 7x^3/3 + 5x^4/4 + 4x); the single-purpose pages sort them by power. An integral ends with + C.
• Solutions lists each real solution once, in increasing order, separated by commas: x = -3, x = 2. An exact solution uses =; a solution with no exact form, or one the algebra gave as a rounded number, uses ≈ and 10 significant figures. With none, it reads no real solution.
• An answer never holds 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, a decimal with more than 12 significant digits, or a whole number past 2⁵³. Such an answer is not shown.
• Value is a decimal to 10 significant figures.

Assumptions

• Angles are in radians; ln and log are the natural logarithm; complex values and solutions are left out.
• An answer that fails its check, finds no formula, or takes over 3 seconds is not shown ("No verified answer").

Worked examples by hand

(3/4 + 1/6) × 12 + √50 (hand calculation). 3/4 + 1/6 = 9/12 + 2/12 = 11/12, and 11/12 × 12 = 11. √50 = √(25 × 2) = 5√2. So the value is 11 + 5√2 ≈ 18.07106781.

Solve 2x + 7 = 19 (OpenStax College Algebra 2e, section 2.2, Example 1). Subtract 7: 2x = 12. Divide by 2: x = 6.

Derivative of 2x⁵ + 7 (OpenStax Calculus Volume 1, section 3.3, Example 3.21). By the power rule the derivative of 2x⁵ is 10x⁴, and the derivative of the constant 7 is 0: 10x⁴.

Integral of 5x³ − 7x² + 3x + 4 (OpenStax Calculus Volume 1, section 4.10, Example 4.52a). Integrate term by term with the power rule, xⁿ → xⁿ⁺¹/(n + 1): 5x⁴/4 − 7x³/3 + 3x²/2 + 4x + C (the page may list the terms in another order).

Simplify (x² − 9)/(x² + 4x + 3) (OpenStax College Algebra 2e, section 1.6, Example 1). (x − 3)(x + 3)/((x + 1)(x + 3)) = (x − 3)/(x + 1).