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Illustration of Newton's method
An example of using Newton–Raphson method lớn solve numerically the equation f(x) = 0

In mathematics, lớn solve an equation is lớn find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values lớn the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The phối of all solutions of an equation is its solution phối.

An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions.

For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible lớn take the variable y lớn be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions lớn the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1).

The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common lớn reserve x, y, z, ... lớn denote the unknowns, and lớn use a, b, c, ... lớn denote the known variables, which are often called parameters. This is typically the case when considering polynomial equations, such as quadratic equations. However, for some problems, all variables may assume either role.

Depending on the context, solving an equation may consist lớn find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging lớn a given interval. When the task is lớn find the solution that is the best under some criterion, this is an optimization problem. Solving an optimization problem is generally not referred lớn as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution.


One general khuông of an equation is

where f is a function, x1, ..., xn are the unknowns, and c is a constant. Its solutions are the elements of the inverse image

where D is the domain name of the function f. The phối of solutions can be the empty phối (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).

For example, an equation such as

with unknowns x, y and z, can be put in the above khuông by subtracting 21z from both sides of the equation, lớn obtain

In this particular case there is not just one solution, but an infinite phối of solutions, which can be written using phối builder notation as

One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the phối of all points whose coordinates are solutions of the equation.

Solution sets[edit]

The solution phối of the equation x2/4 + y2 = 1 forms an ellipse when interpreted as a phối of Cartesian coordinate pairs.

The solution phối of a given phối of equations or inequalities is the phối of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all the equations or inequalities. If the solution phối is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.

For a simple example, consider the equation

This equation can be viewed as a Diophantine equation, that is, an equation for which only integer solutions are sought. In this case, the solution phối is the empty phối, since 2 is not the square of an integer. However, if one searches for real solutions, there are two solutions, 2 and 2; in other words, the solution phối is {2, −2}.

When an equation contains several unknowns, and when one has several equations with more unknowns phàn nàn equations, the solution phối is often infinite. In this case, the solutions cannot be listed. For representing them, a parametrization is often useful, which consists of expressing the solutions in terms of some of the unknowns or auxiliary variables. This is always possible when all the equations are linear.

Such infinite solution sets can naturally be interpreted as geometric shapes such as lines, curves (see picture), planes, and more generally algebraic varieties or manifolds. In particular, algebraic geometry may be viewed as the study of solution sets of algebraic equations.

Methods of solution[edit]

The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns. The variety in types of equations is large, and so sánh are the corresponding methods. Only a few specific types are mentioned below.

In general, given a class of equations, there may be no known systematic method (algorithm) that is guaranteed lớn work. This may be due lớn a lack of mathematical knowledge; some problems were only solved after centuries of effort. But this also reflects that, in general, no such method can exist: some problems are known lớn be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970.

For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology phàn nàn pencil and paper. In some other cases, heuristic methods are known that are often successful but that are not guaranteed lớn lead lớn success.

Brute force, trial and error, inspired guess[edit]

If the solution phối of an equation is restricted lớn a finite phối (as is the case for equations in modular arithmetic, for example), or can be limited lớn a finite number of possibilities (as is the case with some Diophantine equations), the solution phối can be found by brute force, that is, by testing each of the possible values (candidate solutions). It may be the case, though, that the number of possibilities lớn be considered, although finite, is so sánh huge that an exhaustive tìm kiếm is not practically feasible; this is, in fact, a requirement for strong encryption methods.

As with all kinds of problem solving, trial and error may sometimes yield a solution, in particular where the khuông of the equation, or its similarity lớn another equation with a known solution, may lead lớn an "inspired guess" at the solution. If a guess, when tested, fails lớn be a solution, consideration of the way in which it fails may lead lớn a modified guess.

Elementary algebra[edit]

Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as

can be solved using the methods of elementary algebra.

Systems of linear equations[edit]

Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on linear algebra. See Gaussian elimination

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Polynomial equations[edit]

Polynomial equations of degree up lớn four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example

(by using the rational root theorem), and

(by using the substitution x = z13, which simplifies this lớn a quadratic equation in z).

Diophantine equations[edit]

In Diophantine equations the solutions are required lớn be integers. In some cases a brute force approach can be used, as mentioned above. In some other cases, in particular if the equation is in one unknown, it is possible lớn solve the equation for rational-valued unknowns (see Rational root theorem), and then find solutions lớn the Diophantine equation by restricting the solution phối lớn integer-valued solutions. For example, the polynomial equation

has as rational solutions x = −1/2 and x = 3, and so sánh, viewed as a Diophantine equation, it has the unique solution x = 3.

In general, however, Diophantine equations are among the most difficult equations lớn solve.

Inverse functions[edit]

In the simple case of a function of one variable, say, h(x), we can solve an equation of the khuông h(x) = c for some constant c by considering what is known as the inverse function of h.

Given a function h : AB, the inverse function, denoted h−1 and defined as h−1 : BA, is a function such that

Now, if we apply the inverse function lớn both sides of h(x) = c, where c is a constant value in B, we obtain

and we have found the solution lớn the equation. However, depending on the function, the inverse may be difficult lớn be defined, or may not be a function on all of the phối B (only on some subset), and have many values at some point.

If just one solution will tự, instead of the full solution phối, it is actually sufficient if only the functional identity

holds. For example, the projection π1 : R2R defined by π1(x, y) = x has no post-inverse, but it has a pre-inverse π−1
defined by π−1
(x) = (x, 0)
. Indeed, the equation π1(x, y) = c is solved by

Examples of inverse functions include the nth root (inverse of xn); the logarithm (inverse of ax); the inverse trigonometric functions; and Lambert's W function (inverse of xex).


If the left-hand side expression of an equation P = 0 can be factorized as P = QR, the solution phối of the original solution consists of the union of the solution sets of the two equations Q = 0 and R = 0. For example, the equation

can be rewritten, using the identity tan x cot x = 1 as

which can be factorized into

The solutions are thus the solutions of the equation tan x = 1, and are thus the set

Numerical methods[edit]

With more complicated equations in real or complex numbers, simple methods lớn solve equations can fail. Often, root-finding algorithms lượt thích the Newton–Raphson method can be used lớn find a numerical solution lớn an equation, which, for some applications, can be entirely sufficient lớn solve some problem.

Matrix equations[edit]

Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra.

Differential equations[edit]

There is a vast toàn thân of methods for solving various kinds of differential equations, both numerically and analytically. A particular class of problem that can be considered lớn belong here is integration, and the analytic methods for solving this kind of problems are now called symbolic integration.[citation needed] Solutions of differential equations can be implicit or explicit.[1]

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See also[edit]

  • Extraneous and missing solutions
  • Simultaneous equations
  • Equating coefficients
  • Solving the geodesic equations
  • Unification (computer science) — solving equations involving symbolic expressions