Equation Solver

• Type the two sides with one equals sign between them: x^2 - 5*x + 6 = 0. Use + - * /, ^ for powers and parentheses. The multiplication sign may be left out where the meaning is plain: 2x, 3(x+1), (x+1)(x-2).
• Functions need parentheses: sqrt(x), abs(x) or |x|, exp(x) or e^x, ln(x) (natural logarithm), log(x) (base 10), sin(x), cos(x), tan(x), in radians. The constants are pi and e.
• Numbers are read exactly: 0.125 is 1/8. Scientific notation takes a capital E, 1.5E-3, because a small e is always the constant: 2e-3 means 2e − 3.
• Implicit multiplication has the same precedence as * and /, so 1/2x is (1/2)x; write 1/(2x) for the other meaning. The line Reads as under the box always shows how the equation was understood, before anything is solved.
• The unknown is one letter, x unless you choose another. Enter solves. An equation with two different letters is refused, and so is anything the solver cannot read; nothing typed is ever run as code.

• Linear and quadratic equations, with fractions as coefficients. A quadratic is solved from its discriminant, and square roots are simplified: x² − 2 = 0 gives x = ±√2.
• Polynomial equations up to degree 40. Rational roots are found with the Rational Root Theorem and divided out; a quadratic that is left is solved by the formula, and so is a factor of the form xᵏ = c. Any other factor has its real roots counted exactly and then located as decimals.
• Rational equations (quotients of polynomials), with the values that make a denominator zero excluded.
• One square root or one absolute value with the rest polynomial, such as √(x+1) = x − 1 or |2x − 1| = x + 4.
• Equations that undo one operation at a time: 2e^(3x) + 1 = 7, ln(x² − 3) = 0, 2ˣ = 8; and equations in one repeated expression, such as e^(2x) − 3eˣ + 2 = 0.
• sin, cos and tan of a linear expression, such as cos(2x + 1) = 1/2: the general solution, and its members in the interval.

Everything else, for example cos x = x, two square roots, or a sum of logarithms, is solved numerically on the interval. The solver is not a general computer algebra system, and says so rather than guess.

An exact answer is written with =: a fraction, a root, a logarithm, a multiple of π. A decimal that had to be rounded is written with ≈ and is never presented as exact. Precision sets the number of significant digits of the decimals only; it never changes an exact answer.

Decimals of fractions, square roots and other roots of polynomials are expanded with whole-number arithmetic, to as many as 16 digits. Values that pass through ln, eˣ or the trigonometric functions come from ordinary double-precision arithmetic and are shown to at most 15.

Solutions are looked for among the real numbers. So x² + 1 = 0 has no real solutions, although it has two complex ones; that is different from 2x + 1 = 2x + 3, which nothing solves at all.

An equation only makes sense where both sides are defined. A denominator cannot be zero, because division by zero has no value; a square root needs a number that is not negative; a logarithm needs a positive number. The values ruled out are listed as domain restrictions. They stay ruled out even when a factor cancels:

In the same way x/x = 1 holds for every real number except 0.

Some steps are not reversible. Squaring both sides is the usual one: a = b implies a² = b², but a² = b² also holds when a = −b. So the squared equation can have solutions that the original does not: extraneous roots.

For √(x+1) = x − 1, squaring gives x² − 3x = 0, with roots 0 and 3. A square root is never negative, so the right-hand side must not be either: at x = 0 it is −1, and 0 is rejected; at x = 3 it is 2, and 3 is the solution. Rejected candidates are listed with the result, and Verify solutions substitutes every answer into the equation as it was entered.

A root can count more than once: (x − 1)²(x + 2) = 0 has the roots 1 and −2, and 1 has multiplicity 2 because its factor appears twice. The solver lists each solution once, gives the number of distinct solutions, and marks the multiplicity of a repeated root of a polynomial or rational equation.

Most equations with sin, eˣ or ln mixed with powers of x have no solution that can be written with a formula, so decimals are the best there is. The solver then examines f(x) = left − right on the interval from Minimum to Maximum: it evaluates f at 4001 evenly spaced points, and wherever the sign changes it closes in on the root with Brent’s method, which never leaves the enclosing interval. A candidate is kept only if it lies in the domain and the two sides agree there to about nine digits; this is what removes the jumps of tan and the like.

Two limits are worth knowing. Roots closer together than one four-thousandth of the interval can be missed, so a narrower interval looks more closely. And a root where the two sides touch without crossing, like (x − 1)² = 0 would be, is recognised only to about half the digits.

For a trigonometric equation that is solved exactly, the interval only selects which of the infinitely many solutions are listed; the general solution is given as well.

Every result says which of these holds:

• All real solutions were found. This is said for the equations solved exactly. For polynomials it rests on a Sturm sequence, which counts the real roots with exact arithmetic, so even roots given as decimals are known to be all there are.
• The general solution describes every solution, for trigonometric equations, with the list complete inside the interval.
• Numerical solutions found in the interval. Other solutions may exist, in the interval or outside it.
• No solution found in the interval. This is not a proof that there is none.