(* Examples of solving equations/finding roots *)
(* Simply paste each one at the Mathemtica prompt *)

(* Get a symbolic answer. Analytical solution *)

Solve[x^2 - 1 == 0, x]

Solve[ {x^2 + y^2 ==1, x + y ==0 }, {x, y}]


(* Same, but get a numerical value of the answer *)

NSolve[ {x^2 + y^2 ==1, x + y ==0 }, {x, y}]

(* find the sum of the above solutions, x+ y *)
x + y /.  %


(* Good for polynomials, should give you all the roots *)
NSolve[ {x^2 - 1 == 0}, x ]



(* No analytical/symbolic solution *)
Solve[Cos[x] == x, x]

NSolve[Cos[x] == x, x]

(* But you can get an answer using a "full-fledged" (Newton's) numerical method *)
(* You will need to specify a starting point (1 in the example below) *)
(* for the iterations. *) 
FindRoot[Cos[x] == x, {x, 1} ]

(* If there is more than one root, the algorithm will most likely *)
(* converge (if it all ) to a root nearest to the starting point. *) 
(* Try out a few different starting points *)
(* It is always a good idea to plot the function first to see what *)
(* to expect *)

FindRoot[Sin[1/x] == 0.5, {x,  0.1}]