Solve

The list of solutions of an equation, or a system of equations, for its unknown(s).

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageSolve!↗

Solve(equation, unknown)solves equation (an Equal, or a bare expression read as = 0) for unknown.

Solve(equation)as above; the unknown defaults to the equation's single free variable, or to x when there are several and one of them is x.

Solve([eq1, eq2, …], [x, y, …])solves a system of equations; each solution is a tuple of values in the order of the variable list.

Solve(equation, unknown?)The list of solutions of an equation, or a system of equations, for its unknown(s).

Typed signatures

Type	Package	Applies
(any, any*) -> list	analyticreplaces evaluation	always
(any, any*) -> list	compute-engine	always
(any, any*) -> list	evaluationreplaces compute-engine	always

Domain: Compute engine

Examples

3 more kept as data, checked against the oracles

Basic #

Solve $x^2 = 1$ for $x$

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In$\mathrm{Solve}(x^2=1, x)$Out$\bigl\lbrack1, -1\bigr\rbrack$

Solve $x^2 - 4$; a bare expression reads as $= 0$

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In$\mathrm{Solve}(x^2-4, x)$Out$\bigl\lbrack2, -2\bigr\rbrack$

Solve $x + 1 = 0$, a single real root

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In$\mathrm{Solve}(x+1=0, x)$Out$\bigl\lbrack-1\bigr\rbrack$

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In$\mathrm{Solve}(x^2+ax+1=0, x)$Out$\bigl\lbrack\frac{1}{2}(\sqrt{a^2-4}-a), \frac{1}{2}(-a-\sqrt{a^2-4})\bigr\rbrack$

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In$\mathrm{Solve}((x^2+2)(x^2-2)=0, x∈ℝ)$Out$\bigl\lbrack-\sqrt{2}, \sqrt{2}\bigr\rbrack$

Scope #

Solve $x^2 + 1 = 0$; yields complex roots

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In$\mathrm{Solve}(x^2+1=0, x)$Out$\bigl\lbrack i, -i\bigr\rbrack$

Contradictory equations have no solution: an empty list

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In$\mathrm{Solve}(x=1∧x=2, x)$Out$\bigl\lbrack\bigr\rbrack$

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In$\mathrm{Solve}(ax^2+bx+c=0, x)$Out$\bigl\lbrack\frac{\sqrt{b^2-4ac}-b}{2a}, \frac{-b-\sqrt{b^2-4ac}}{2a}\bigr\rbrack$

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In$\mathrm{Solve}((x^4-1)(x^4-4)=0, x∈ℂ)$Out$\bigl\lbrack-\sqrt{2}, -1, 1, \sqrt{2}, \sqrt{2}i, i, -i, -\sqrt{2}i\bigr\rbrack$

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In$\mathrm{Solve}((x^4-1)(x^4-4)=0, x∈ℝ)$Out$\bigl\lbrack-\sqrt{2}, -1, 1, \sqrt{2}\bigr\rbrack$

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In$\mathrm{Solve}((x^4-1)(x^4-4)=0, x∈ℤ)$Out$\bigl\lbrack-1, 1\bigr\rbrack$

Over the reals, $x^2 = a$ has roots only for $a \ge 0$; with $a$ unknown the call stays unevaluated

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In$\mathrm{Solve}(x^2=a, x∈ℝ)$Out$\mathrm{Solve}(x^2=a, x∈ℝ)$

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In$\mathrm{Solve}(x^4-x^2-5=0, x)$Out$\bigl\lbrack-\sqrt{\frac{1}{2}+\frac{21^{1/2}}{2}}, \sqrt{\frac{1}{2}+\frac{\sqrt{21}}{2}}, \sqrt{\frac{1}{2}-\frac{21^{1/2}}{2}}, -\sqrt{1/2-21^{1/2}/2}\bigr\rbrack$

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In$\mathrm{Solve}((x^2-1)(x^4-1)=0, x)$Out$\bigl\lbrack-1, 1, i, -i\bigr\rbrack$

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In$\mathrm{Solve}(ax^5+bx+c=0, x)$Out$\mathrm{Solve}(ax^5+bx+c=0, x)$

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In$\mathrm{Solve}(\sin(x)=\frac{1}{3}, x)$Out$\mathrm{Solve}(\sin(x)=\frac{1}{3}, x)$

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In$\mathrm{Solve}(x^{2a}+2x^{a}+1=0, x)$Out$\mathrm{Solve}(x^{2a}+2x^{a}+1=0, x)$

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In$\mathrm{Solve}((x^5-1)^{x}=0, x)$Out$\mathrm{Solve}((x^5-1)^{x}=0, x)$

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In$\mathrm{Solve}(5=x×2^{x^2}, x)$Out$\mathrm{Solve}(5=x×2^{x^2}, x)$

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In$\mathrm{Solve}(Γ(x)-\ln(x)=\frac{i}{2}∧\vert x-2\vert\lt\frac{3}{2}, x)$Out$\mathrm{Solve}(Γ(x)-\ln(x)=\frac{i}{2}∧\vert x-2\vert\lt\frac{3}{2}, x)$

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In$\mathrm{Solve}(x=e^{1/x}∧\vert x\vert\lt5, x)$Out$\mathrm{Solve}(x=e^{1/x}∧\vert x\vert\lt5, x)$

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In$\mathrm{Solve}(\cos(e^{x})=3e^{-x}+1∧0≤\mathrm{Re}(x)≤1, x)$Out$\mathrm{Solve}(\cos(e^{x})=3e^{-x}+1∧0≤\mathrm{Re}(x)≤1, x)$

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In$\mathrm{Solve}(\begin{cases}x+2y+3z=4\\3x+4y+5z=6\\7x+7y+8z=9\end{cases}, x, y, z)$Out$\bigl\lbrack(0,-1,2)\bigr\rbrack$

Applications #

A 2x2 linear system; each solution is a tuple of $(x, y)$

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In$\mathrm{Solve}(\begin{cases}x+y=3\\x-y=1\end{cases}, \bigl\lbrack x, y\bigr\rbrack)$Out$\bigl\lbrack(2,1)\bigr\rbrack$

About

Examples	28
Checked against	Wolfram Language 17 agree11 differ
Crosswalk	Wolfram Language

See also: Root, D