Solve equations

Solving equations walks through the first two; this page is the summary.

One equation

Write	Gives
`root(f, x)`	A value of x where f is 0, starting from x’s current value
`root(f, x, a, b)`	A root between a and b (f must change sign there)

x := 1
root(x^3 - 2*x - 5, x) = ?
root(cos(x) - x, x, 0, 1) = ? — As text
x := 1 root(x^3 - 2*x - 5, x) = ? root(cos(x) - x, x, 0, 1) = ?
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If root reports that it found no root near the guess, give limits, or a better guess. It looks for real roots only.

Polynomials

polyroots(v) gives every root of v₀ + v₁x + v₂x² + …, complex ones too:

polyroots([-6; 11; -6; 1]) = ?
polyroots([1; 0; 1]) = ? — As text
polyroots([-6; 11; -6; 1]) = ? polyroots([1; 0; 1]) = ?
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Several equations

a := 1 m
b := 4 m
given
a + b == 5 m
a*b == 6 m^2
find(a, b) = ? — As text
a := 1 m b := 4 m given a + b == 5 m a*b == 6 m^2 find(a, b) = ?
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Every unknown needs a guess above the block, with its unit.
Below given (type given and a space) go the conditions: == (Ctrl+=), <, >, <=, >=. A chain such as 0 m < L < 10 m is two conditions.
The block ends at the statement that calls find, minerr, Minimize or Maximize.
[a_1; b_1] := find(a, b) names each result.

When there is no exact solution, minerr gives the values that come closest (a least-squares fit), where find reports an error:

x_d := [1; 2; 3; 4]
y_d := [2.1; 3.9; 6.2; 7.8]
a := 1
b := 0
given
a*x_d[0] + b == y_d[0]
a*x_d[1] + b == y_d[1]
a*x_d[2] + b == y_d[2]
a*x_d[3] + b == y_d[3]
minerr(a, b) = ? — As text
x_d := [1; 2; 3; 4] y_d := [2.1; 3.9; 6.2; 7.8] a := 1 b := 0 given a*x_d[0] + b == y_d[0] a*x_d[1] + b == y_d[1] a*x_d[2] + b == y_d[2] a*x_d[3] + b == y_d[3] minerr(a, b) = ?
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The best values

Minimize(f, x, y, …) and Maximize(f, x, y, …) give the values of the variables where the function f is least or greatest, starting from their current values. Below a given, the conditions hold too:

f(x) := (x - 2)^2 + 1
x := 0
Minimize(f, x) = ? — As text
f(x) := (x - 2)^2 + 1 x := 0 Minimize(f, x) = ?
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w := 1 m
h := 1 m
A(w, h) := w*h
given
2*w + 2*h == 4 m
Maximize(A, w, h) = ? — As text
w := 1 m h := 1 m A(w, h) := w*h given 2*w + 2*h == 4 m Maximize(A, w, h) = ?
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Good guesses

All of these search from the guesses, so the guesses choose the answer when there are several. A guess right between two answers, or exactly on a symmetry of the problem, can leave the search stuck. If a result is surprising, plot the function first and guess near the answer you want.

Functions for solving has the full list.