Solving equations

Package: SYMBOLIC_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

Solving an equation means finding the values of the unknown that make it true: the x for which 2x+6=0, or the two roots of x^2-4=0. The engine gives an exact answer wherever one exists, a square root rather than a decimal, and falls back to accurate decimals only when no exact form can be written. Like the other algebra forms, this does not need a trailing arrow.

solve takes an equation and the unknown to solve for.

solve(2x+6=0, x) // -3
solve(x^2-4=0, x) // [-2, 2]
solve(x^2-3x+2=0, x) // [1, 2]
solve(3x-1=0, x) // 1/3

A missing right-hand side means zero, so solve(x^2-4, x) asks the same question.

solve(x^2-4, x) // [-2, 2]

Writing the equation on its own line

An equation containing exactly one unknown can be written plainly, then solved by asking for that unknown with an arrow. This is the same solver, reached a different way.

x^2-4 = 0
x => // [-2, 2]

2x+1 = x+4
x => // 3

The unknown is whichever name has no value yet, so an equation can refer to variables already defined above it.

:a = 2
a*n = 10
n => // 5

An equation with two unknowns is not stored, because there would be no way to tell which one to solve for. Use solve and name it.

Exact answers, including irrational ones

An irrational root is given as a square root rather than a decimal, in lowest form.

solve(x^2-2=0, x) // [-sqrt(2), sqrt(2)]

Roots that are not rational and not expressible this way are approximated, and only after every exact method has been tried.

A constant term of zero is not a special case, though it used to behave like one. x divides out first, and what is left goes through the exact methods like any other equation.

solve(x^3-x=0, x) // [-1, 0, 1]
solve(x^5-x=0, x) // [-1, 0, 1, -i, i]

That second one is exact all the way through: x^5-x is x(x-1)(x+1)(x^2+1), so all five roots are a rational or the imaginary unit and no approximation is involved anywhere.

Cubics and quartics

Both have closed forms, and both are used. A cubic goes through Cardano’s method, which returns all three roots including the complex pair.

solve(x^3-8=0, x) // [2, -1-sqrt(3)*i, -1+sqrt(3)*i]
solve(x^3-2=0, x) // [cbrt(2), -cbrt(2)/2-cbrt(2)*sqrt(3)/2*i, -cbrt(2)/2+cbrt(2)*sqrt(3)/2*i]

Cardano’s formula for a cubic that has no neat root produces a genuinely long expression, nested cube roots over square roots. It is returned anyway, because it is the exact answer and rounding it away would throw information out.

A quartic is solved when it has no odd power, or when it splits into two quadratics with rational coefficients.

solve(x^4-4x^2+4=0, x) // [-sqrt(2), sqrt(2)]
solve(x^4-3x^2+1=0, x) // [-sqrt((3+sqrt(5))/2), sqrt((3+sqrt(5))/2), -sqrt((3-sqrt(5))/2), sqrt((3-sqrt(5))/2)]
solve(x^4+4x^2+4x+15=0, x) // [-1-sqrt(2)*i, -1+sqrt(2)*i, 1-2i, 1+2i]

A biquadratic whose roots are complex is still exact, nesting one square root inside another where it has to.

solve(x^4-2x^2+3=0, x) // [-sqrt((sqrt(3)+1)/2)-sqrt((sqrt(3)-1)/2)*i, -sqrt((sqrt(3)+1)/2)+sqrt((sqrt(3)-1)/2)*i, sqrt((sqrt(3)+1)/2)-sqrt((sqrt(3)-1)/2)*i, sqrt((sqrt(3)+1)/2)+sqrt((sqrt(3)-1)/2)*i]

What stays approximate

A cubic with three distinct real roots and no rational one is the casus irreducibilis, and its roots are reported as decimals.

solve(x^3-3x+1=0, x) // [-1.88, 0.35, 1.53]

That is not a gap in effort. It is a theorem that those three roots cannot be written with real radicals at all: Cardano’s formula reaches them only by taking cube roots of complex numbers, and the expression that does exist involves a cosine of an arccosine of an irrational. Three accurate decimals are the more useful answer.

Any remaining quartic, and every equation of degree five to eight, is solved numerically for the same kind of reason. A general quartic does have a closed form, four radicals deep, and no one can read it. A quintic has none at all, which is Abel’s theorem rather than a limitation of this solver.

The numerical method finds every root at once, in the complex plane, so what comes back is the whole set and not the part of it that happens to lie on the real line.

solve(x^5-1=0, x) // [1, -0.8090169944-0.5877852523i, -0.8090169944+0.5877852523i, 0.3090169944-0.9510565163i, 0.3090169944+0.9510565163i]

A partial answer is never given as a whole one

The number of roots an equation has is its degree, and the solver knows that before it starts. If it reaches the end with fewer, it reports how many are missing rather than handing back the ones it found, because a short list of correct roots reads exactly like a complete one and there is nothing in it for a reader to notice.

Answers that are not numbers

Some equations have a correct answer that is not a list of roots.

solve(1=2, x) // no solution

x^2+1=0 does have solutions, and they are complex. Solving in terms of another unknown works when the equation is linear in the one being solved for.

solve(a*x+b=0, x) // -b/a

If you do not want these words

factor, solve, expand, der, derivative, integral, taylor and jacobian are only treated as functions when the very next character is an opening parenthesis, so they remain usable as ordinary variable names.

:factor = 1.5
:factor * 2 // 3

If a host wants them gone entirely, the package can be left out at registration. That is a decision about which words your grammar claims, not a performance one: it does not measurably speed the engine up, and it does not make the bundle smaller. See the package system for the measurements.