The Roots(f(x), x) command, where f(x) is an algebraic expression, returns as a list of values points where f(x) is zero.

The Roots(f(x), x = a..b) command, where f(x) is an algebraic expression, returns as a list of values points where f(x) is zero in the interval [a,b].

If f(x) is an equation, this routine finds the roots of the expression $\mathrm{lhs}\left(f\left(x\right)\right)-\mathrm{rhs}\left(f\left(x\right)\right)$, in the specified interval if one is provided.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

A root (or zero) of f(x) is defined as any point at which the value of the expression f(x) is $0$ (zero).

If the expression has an infinite number of roots, a warning message and sample roots are returned.

The opts argument can contain the following equation that sets computation options.

numeric = true or false

Whether to use numeric methods (using floating-point computations) to find the roots of the expression. If this option is set to true, the points a and b must be finite and are set to $\mathrm{-10}$ and $10$ if they are not provided. The default value of this option is determined by the numeric values in the expression $f\left(x\right)$: If any of these numeric values is a floating point number (has a decimal point), then this option defaults to to true.  Note: If the numeric option is in effect, then the command does not warn about an infinite number of roots.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

$\mathrm{Roots}\left(3x^2-x\,x\right)$

$\left[0\,\frac{1}{3}\right]$

$\mathrm{Roots}\left(\sin \left(x\right)\,0..10\right)$

$\left[0\,\mathrm{\pi }\,2\mathrm{\pi }\,3\mathrm{\pi }\right]$

$\mathrm{Roots}\left(\sin \left(\frac{1.0}{x}\right)\,-1..1\right)$

$\left[\mathrm{-0.3183098862}\,\mathrm{-0.1591549431}\,\mathrm{-0.1061032954}\,\mathrm{-0.07957747155}\,\mathrm{-0.06366197724}\,\mathrm{-0.05305164770}\,\mathrm{-0.03978873577}\,\mathrm{-0.02448537586}\,\mathrm{-0.009947183943}\,0.004973591972\,0.01989436789\,0.03536776513\,0.05305164770\,0.06366197724\,0.07957747155\,0.1061032954\,0.1591549431\,0.3183098862\right]$

$\mathrm{Roots}\left(2x^3+5x^2=4x\,x\right)$

$\left[-\frac{5}{4}-\frac{\sqrt{57}}{4}\,0\,-\frac{5}{4}+\frac{\sqrt{57}}{4}\right]$

$\mathrm{Roots}\left(\frac{x^2-3x+1}{x}\right)$

$\left[\frac{3}{2}-\frac{\sqrt{5}}{2}\,\frac{3}{2}+\frac{\sqrt{5}}{2}\right]$

$\mathrm{Roots}\left(\frac{x^2-3x+1}{x}=0\,\mathrm{numeric}\right)$

$\left[0.3819660113\,2.618033989\right]$