`Solve`¶

Symbolic equation solver.

$ wo 'Solve[x^2 == 4, x]'
{{x -> -2}, {x -> 2}}

$ wo 'Solve[{x + y == 3, x - y == 1}, {x, y}]'
{{x -> 2, y -> 1}}

A negative leading coefficient still gives simplified, correctly ordered roots.

$ wo 'Solve[2 - x^2 == 0, x]'
{{x -> -Sqrt[2]}, {x -> Sqrt[2]}}

$ wo 'Solve[6 - x - x^2 == 0, x]'
{{x -> -3}, {x -> 2}}

x^n == c is solved in radicals, the roots being the real root times the n-th roots of unity:

$ wo 'Solve[x^3 == 8, x]'
{{x -> 2}, {x -> -2*(-1)^(1/3)}, {x -> 2*(-1)^(2/3)}}

$ wo 'Solve[x^3 == 2, x]'
{{x -> -(-2)^(1/3)}, {x -> 2^(1/3)}, {x -> (-1)^(2/3)*2^(1/3)}}

For an odd degree and a negative right-hand side the generating root is the real one:

$ wo 'Solve[x^3 == -8, x]'
{{x -> -2}, {x -> 2*(-1)^(1/3)}, {x -> -2*(-1)^(2/3)}}

NSolve stays on the numeric root finder, so a conjugate pair agrees to the last bit:

$ wo 'NSolve[x^3 == 8, x]'
{{x -> -1. - 1.7320508075688772*I}, {x -> -1. + 1.7320508075688772*I}, {x -> 2.}}

Inverting Abs splits the equation into a positive and a negative branch, and over the complexes that leaves out the rest of the circle — Solve says so:

$ wo 'Solve[Abs[x] == 2, x]'

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
{{x -> -2}, {x -> 2}}

Restricting the domain to the reals, or narrowing the split with a constraint, loses nothing and so reports nothing:

$ wo 'Solve[Abs[x] == 2, x, Reals]'
{{x -> -2}, {x -> 2}}

$ wo 'Solve[{Abs[x] == 2, x > 0}, x]'
{{x -> 2}}

A list of constraints is the same system as the conjunction of them:

$ wo 'Solve[{Sin[x] == 0, 0 < x < 7}, x]'
{{x -> Pi}, {x -> 2*Pi}}

Modulus -> n solves over the integers modulo n, so the answers are residues rather than radicals:

$ wo 'Solve[x^2 == 2, x, Modulus -> 7]'
{{x -> 3}, {x -> 4}}

$ wo 'Solve[x^3 == 1, x, Modulus -> 7]'
{{x -> 1}, {x -> 2}, {x -> 4}}

There may be none, and the modulus need not be prime:

$ wo 'Solve[x^2 == 3, x, Modulus -> 7]'
{}

$ wo 'Solve[2 x == 4, x, Modulus -> 6]'
{{x -> 2}, {x -> 5}}

MaxRoots -> n keeps only the first n solutions:

$ wo 'Solve[x^4 == 1, x, MaxRoots -> 3]'
{{x -> -1}, {x -> -I}, {x -> I}}

It has to be a positive integer, Infinity or Automatic:

$ wo 'Solve[x^3 == 1, x, MaxRoots -> 0]'

Solve::maxrts: The value 0 of the MaxRoots option is not a positive integer, Infinity or Automatic.
Solve[x^3 == 1, x, MaxRoots -> 0]

Machine-precision coefficients are solved numerically instead of in radicals or as Root objects:

$ wo 'Solve[x^3 == 8., x]'
{{x -> -1. - 1.7320508075688772*I}, {x -> -1. + 1.7320508075688772*I}, {x -> 2.}}

$ wo 'Round[x /. Solve[x^3 + 1.5 x^2 - 3.2 x + 4.7 == 0, x], 1/10^6]'
{-19079/6250, 2426/3125 - (120997*I)/125000, 2426/3125 + (120997*I)/125000}

Every root is reported with its multiplicity:

$ wo 'Solve[x^3 - 4 x^2 == 0, x]'
{{x -> 0}, {x -> 0}, {x -> 4}}

An inequality alongside the equation keeps only the roots inside it — including roots with no radical form, which are decided on their value:

$ wo 'Solve[x^4 == 16 && x > 0, x]'
{{x -> 2}}

$ wo 'Solve[1 == f^2 (3 - f) && 0 <= f <= 1, f]'
{{f -> Root[1 - 3*#1^2 + #1^3 & , 2, 0]}}

An equation with a list on one side threads over that list, so a scalar on the other side is compared against every element:

$ wo 'Solve[{x - 1, y - 2} == 0, {x, y}]'
{{x -> 1, y -> 2}}

That makes Solve[Table[…] == 0] work, with the variable inferred from the equations:

$ wo 'Solve[Table[BernoulliB[n, z], {n, 3, 3}] == 0]'
{{z -> 0}, {z -> 1/2}, {z -> 1}}

Solve¶

`Solve`¶