Root finding in Mathematica

Question

I am new to Mathematica. I have used the WolframAlpha online calculator to find roots of equations (listed under the heading Root in the output generated in response to a submission). When I try to find the roots of the same equation in Mathematica, I receive various errors. I have tried FindRoot and Reduce. Is there a function in Mathematica that corresponds to whatever is generating the Root output in the online WolframAlpha calculator? Thank you.

Answer 1

The canonical way to solve algebraic equations is to use Solve, e.g.

2 - 4 x - x^2 + 2 x^3 == 0 // Solve[#, x] &

{{x -> 1/2}, {x -> -Sqrt[2]}, {x -> Sqrt[2]}}

The above eqn //Solve[#,x]& is a so called postfix notation.

This is an infix notation: eqn ~ Solve ~ x

and prefix : Solve[ #, x]& @(eqn)

If you want to solve a transcendental equation e.g. x^2 == Cos[x] you can use Solve as well as Reduce, however one should specify the real domain because by default Mathematica working in the complex domain cannot find all solutions.

Reduce[x^2 == Cos[x], x, Reals]

x == Root[{-Cos[#1] + #1^2 &, -0.82413231230252242296}] ||
x == Root[{-Cos[#1] + #1^2 &, 0.82413231230252242296}]

Solve[x^2 == Cos[x], x, Reals]

{{x -> Root[{-Cos[#1] + #1^2 &, -0.82413231230252242296}]},
 {x -> Root[{-Cos[#1] + #1^2 &, 0.82413231230252242296}]}}

If you are interested in numerical values of solutions you could also choose FindRoot, it works like this :

 FindRoot[ x^2 == Cos[x], {x, Pi/6}]

{x -> 0.824132}

In general x0, here Pi/6 is the point where it starts to search a numerical solution.

NSolve also tackles equations numerically:

NSolve[x^2 == Cos[x], x, Reals]

{{x -> -0.824132}, {x -> 0.824132}}

And here I briefly sketch a few samples of using functions which you mentioned :

Roots[ 2 - 4 x - x^2 + 2 x^3 == 0, x]

x == Sqrt[2] || x == -Sqrt[2] || x == 1/2

Root[ 2 - 4 x - x^2 + 2 x^3, #] & /@ Range[3]

{-Sqrt[2], 1/2, Sqrt[2]}

Reduce[ 2 - 4 x - x^2 + 2 x^3== 0, x]

 x == 1/2 || x == -Sqrt[2] || x == Sqrt[2]

Answer 2

Solve may be a good start:

Solve[a x^2 + b x + c == 0, x]

{
    {x -> (-b - Sqrt[b^2 - 4 a c])/(2 a)},
    {x -> (-b + Sqrt[b^2 - 4 a c])/(2 a)}
}

Other functions that are often useful when solving equations are Reduce (find a logical expression for all variables satisfying a statement) and Refine (add constraints on variables, such as being positive)

FindRoot, one of the functions you used, solves numerically only, iterating Newton's method until sufficiently accurate.

Root is, as far as I know, not really meant to find roots, it's much more an abstract representation of the $n$th root of an equation, much like $\sin(10)$ does make sense on its own, without evaluating it to an explcit number. In some cases for example, it may be impossible to calculate the root of an equation explicitly, so Mathematica keeps it as a Root object; however, once you apply numerical functions like N to it, it can still do some useful operations with it.