# Equations in One Variable

The main equations that Solve and related *Mathematica* functions deal with are *polynomial equations*.

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*Mathematica*can also find exact solutions to cubic equations. Here is the first solution to a comparatively simple cubic equation.

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For cubic and quartic equations the results are often complicated, but for all equations with degrees up to four *Mathematica* is always able to give explicit formulas for the solutions.

An important feature of these formulas is that they involve only *radicals*: arithmetic combinations of square roots, cube roots and higher roots.

It is a fundamental mathematical fact, however, that for equations of degree five or higher, it is no longer possible in general to give explicit formulas for solutions in terms of radicals.

There are some specific equations for which this is still possible, but in the vast majority of cases it is not.

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Root[f,k] | the k root of the equation |

Implicit representation for roots.

*Mathematica*uses an implicit symbolic representation.

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Root objects provide an exact, though implicit, representation for the roots of a polynomial. You can work with them much as you would work with Sqrt[2] or any other expression that represents an exact numerical quantity.

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If the only symbolic parameter that exists in an equation is the variable that you are solving for, then all the solutions to the equation will just be numbers. But if there are other symbolic parameters in the equation, then the solutions will typically be functions of these parameters.

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If you give Solve any -degree polynomial equation, then it will always return exactly solutions, although some of these may be represented by Root objects. If there are degenerate solutions, then the number of times that each particular solution appears will be equal to its multiplicity.

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*Mathematica* also knows how to solve equations which are not explicitly in the form of polynomials.

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So long as it can reduce an equation to some kind of polynomial form, *Mathematica* will always be able to represent its solution in terms of Root objects. However, with more general equations, involving, say, transcendental functions, there is no systematic way to use Root objects, or even necessarily to find numerical approximations.

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Polynomial equations in one variable only ever have a finite number of solutions. But transcendental equations often have an infinite number. Typically the reason for this is that functions like Sin in effect have infinitely many possible inverses. With the default option setting InverseFunctions->True, Solve will nevertheless assume that there is a definite inverse for any such function. Solve may then be able to return particular solutions in terms of this inverse function.

*Mathematica*returns a particular solution in terms of ArcSin, but prints a warning indicating that other solutions are lost.

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If you ask Solve to solve an equation involving an arbitrary function like , it will by default try to construct a formal solution in terms of inverse functions.

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InverseFunction[f] | the inverse function of f |

InverseFunction[f,k,n] | the inverse function of the n-argument function f with respect to its k argument |

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While Solve can only give specific solutions to an equation, Reduce can give a representation of a whole solution set. For transcendental equations, it often ends up introducing new parameters, say with values ranging over all possible integers.

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As discussed at more length in "Equations and Inequalities over Domains", Reduce allows you to restrict the domains of variables. Sometimes this will let you generate definite solutions to transcendental equations—or show that they do not exist.

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