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Student[Calculus1]

 Roots
 find the real roots (zeros) of an expression

 Calling Sequence Roots(f(x), x, opts) Roots(f(x), x = a..b, opts) Roots(f(x), a..b, opts)

Parameters

 f(x) - algebraic expression or equation in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify restricted interval for roots opts - equation(s) of the form numeric=true or false; specify computation options

Description

 • 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 $-10$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):$
 > $\mathrm{Roots}\left(3{x}^{2}-x,x\right)$
 $\left[{0}{,}\frac{{1}}{{3}}\right]$ (1)
 > $\mathrm{Roots}\left(\mathrm{sin}\left(x\right),0..10\right)$
 $\left[{0}{,}{\mathrm{π}}{,}{2}{}{\mathrm{π}}{,}{3}{}{\mathrm{π}}\right]$ (2)
 > $\mathrm{Roots}\left(\mathrm{sin}\left(\frac{1.0}{x}\right),-1..1\right)$
 $\left[{-}{0.3183098862}{,}{-}{0.1591549431}{,}{-}{0.1061032954}{,}{-}{0.07957747155}{,}{-}{0.06366197724}{,}{-}{0.05305164770}{,}{-}{0.03978873577}{,}{-}{0.02448537586}{,}{-}{0.009947183943}{,}{0.004973591972}{,}{0.01989436789}{,}{0.03536776513}{,}{0.05305164770}{,}{0.06366197724}{,}{0.07957747155}{,}{0.1061032954}{,}{0.1591549431}{,}{0.3183098862}\right]$ (3)
 > $\mathrm{Roots}\left(2{x}^{3}+5{x}^{2}=4x,x\right)$
 $\left[{-}\frac{{5}}{{4}}{-}\frac{{1}}{{4}}{}\sqrt{{57}}{,}{0}{,}{-}\frac{{5}}{{4}}{+}\frac{{1}}{{4}}{}\sqrt{{57}}\right]$ (4)
 > $\mathrm{Roots}\left(\frac{{x}^{2}-3x+1}{x}\right)$
 $\left[\frac{{3}}{{2}}{-}\frac{{1}}{{2}}{}\sqrt{{5}}{,}\frac{{3}}{{2}}{+}\frac{{1}}{{2}}{}\sqrt{{5}}\right]$ (5)
 > $\mathrm{Roots}\left(\frac{{x}^{2}-3x+1}{x}=0,\mathrm{numeric}\right)$
 $\left[{0.3819660113}{,}{2.618033989}\right]$ (6)