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Roots
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Roots

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Roots[lhs==rhs,var]

yields a disjunction of equations which represent the roots of a polynomial equation.

Details and Options

  • Roots uses Factor and Decompose in trying to find roots.
  • You can find numerical values of the roots by applying N.
  • Roots can take the following options:
  • Cubics Truewhether to generate explicit solutions for cubics
    EquatedTo Nullexpression to which the variable solved for should be equated
    Modulus 0integer modulus
    Multiplicity 1multiplicity in final list of solutions
    Quartics Truewhether to generate explicit solutions for quartics
    Using Truesubsidiary equations to be solved
  • Roots is generated when Solve and related functions cannot produce explicit solutions. Options are often given in such cases.
  • Roots gives several identical equations when roots with multiplicity greater than one occur.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Find roots of univariate polynomial equations:

In[1]:=1
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https://wolfram.com/xid/0mre2w-dmsaxi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mre2w-d0tr6a
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

Equation with exact numeric coefficients:

In[1]:=1
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https://wolfram.com/xid/0mre2w-c36bc1
Out[1]=1

Equation with symbolic coefficients:

In[1]:=1
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https://wolfram.com/xid/0mre2w-bfh9n
Out[1]=1

General equations of degree five and higher cannot be solved in radicals:

In[1]:=1
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https://wolfram.com/xid/0mre2w-ijc8hk
Out[1]=1

This equation of degree nine is solved in radicals using factorization and decomposition:

In[1]:=1
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https://wolfram.com/xid/0mre2w-cpuks
Out[1]=1

An equation with inexact numeric coefficients:

In[1]:=1
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https://wolfram.com/xid/0mre2w-d0ay5v
Out[1]=1

Multiple roots are repeated the corresponding number of times:

In[1]:=1
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https://wolfram.com/xid/0mre2w-feh2n7
Out[1]=1

Find roots over the integers modulo 7:

In[1]:=1
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https://wolfram.com/xid/0mre2w-kv2wpo
Out[1]=1

Options  (10)Common values & functionality for each option

Cubics  (3)

By default Roots uses the general formulas for solving cubic equations in radicals:

In[1]:=1
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https://wolfram.com/xid/0mre2w-gf9nqd
Out[1]=1

With Cubics->False, Roots does not use the general formulas for solving cubics in radicals:

In[1]:=1
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https://wolfram.com/xid/0mre2w-c18er5
Out[1]=1

Solving this cubic equation in radicals does not require the general formulas:

In[1]:=1
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https://wolfram.com/xid/0mre2w-esc1ym
Out[1]=1

EquatedTo  (1)

Use EquatedTo to specify the left-hand side of the returned equations:

In[1]:=1
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https://wolfram.com/xid/0mre2w-f1otjv
Out[1]=1

Modulus  (1)

Find roots over the integers modulo 12:

In[1]:=1
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https://wolfram.com/xid/0mre2w-cjhrrw
Out[1]=1

Multiplicity  (1)

With Multiplicity->n, the multiplicity of each root is multiplied by n:

In[1]:=1
✖
https://wolfram.com/xid/0mre2w-ifk0f8
Out[1]=1

Quartics  (3)

By default Roots uses the general formulas for solving quartic equations in radicals:

In[1]:=1
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https://wolfram.com/xid/0mre2w-gl14nz
Out[1]=1

With Quartics->False, Roots does not use the general formulas for solving quartics:

In[1]:=1
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https://wolfram.com/xid/0mre2w-hq2az3
Out[1]=1

Solving this quartic equation in radicals does not require the general formulas:

In[1]:=1
✖
https://wolfram.com/xid/0mre2w-kwu5f
Out[1]=1

Using  (1)

Specify equations satisfied by symbolic parameters:

In[1]:=1
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https://wolfram.com/xid/0mre2w-jf3078
Out[1]=1

Properties & Relations  (5)Properties of the function, and connections to other functions

Solutions returned by Roots satisfy the equation:

In[1]:=1
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https://wolfram.com/xid/0mre2w-ez7fy2
Out[1]=1

Use ToRules to convert equations returned by Roots to replacement rules:

In[2]:=2
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https://wolfram.com/xid/0mre2w-ch7c6p
Out[2]=2

Solve uses Roots to find solutions of univariate equations and returns replacement rules:

In[3]:=3
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https://wolfram.com/xid/0mre2w-kzmu7w
Out[3]=3

Roots finds all complex solutions:

In[1]:=1
✖
https://wolfram.com/xid/0mre2w-k2fbay
Out[1]=1

Use Reduce to find solutions over specified domains:

In[2]:=2
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https://wolfram.com/xid/0mre2w-god7q2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0mre2w-buhzuw
Out[3]=3

Use FindInstance to find one solution:

In[4]:=4
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https://wolfram.com/xid/0mre2w-hys4hs
Out[4]=4

Use Solve or Reduce to find solutions of systems of multivariate equations:

In[1]:=1
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https://wolfram.com/xid/0mre2w-df190g
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0mre2w-ew1sm
Out[2]=2

Use Reduce to find solutions of systems of equations and inequalities:

In[1]:=1
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https://wolfram.com/xid/0mre2w-bxmybn
Out[1]=1

Use NRoots to find numeric approximations of roots of a univariate equation:

In[1]:=1
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https://wolfram.com/xid/0mre2w-etsfhm
Out[1]=1
Wolfram Research (1988), Roots, Wolfram Language function, https://reference.wolfram.com/language/ref/Roots.html.
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Wolfram Research (1988), Roots, Wolfram Language function, https://reference.wolfram.com/language/ref/Roots.html.

Text

Wolfram Research (1988), Roots, Wolfram Language function, https://reference.wolfram.com/language/ref/Roots.html.

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Wolfram Research (1988), Roots, Wolfram Language function, https://reference.wolfram.com/language/ref/Roots.html.

CMS

Wolfram Language. 1988. "Roots." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Roots.html.

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Wolfram Language. 1988. "Roots." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Roots.html.

APA

Wolfram Language. (1988). Roots. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Roots.html

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Wolfram Language. (1988). Roots. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Roots.html

BibTeX

@misc{reference.wolfram_2025_roots, author="Wolfram Research", title="{Roots}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Roots.html}", note=[Accessed: 09-June-2025 ]}

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@misc{reference.wolfram_2025_roots, author="Wolfram Research", title="{Roots}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Roots.html}", note=[Accessed: 09-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_roots, organization={Wolfram Research}, title={Roots}, year={1988}, url={https://reference.wolfram.com/language/ref/Roots.html}, note=[Accessed: 09-June-2025 ]}

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@online{reference.wolfram_2025_roots, organization={Wolfram Research}, title={Roots}, year={1988}, url={https://reference.wolfram.com/language/ref/Roots.html}, note=[Accessed: 09-June-2025 ]}