(* Задача 1. *)
f[x_]:=(x^2)-4x+1;
Solve[f[x]==0,x]

N[((((Sqrt[2-(Sqrt[3])])+(Sqrt[2+(Sqrt[3])])))^(-4))]

(*
Проверка;
1. (Subscript[x, 1])+(Subscript[x, 2]) = ((-b)/a) = ((-(-4))/1) = 4;
2. (Subscript[x, 1])(Subscript[x, 2]) = (c/a) = (1/1) = 1;
=> от 1. и 2. => 2 положителни корена;

(Sqrt[(Subscript[x, 1])])+(Sqrt[(Subscript[x, 2])]) = ?;
((((Sqrt[(Subscript[x, 1])])+(Sqrt[(Subscript[x, 2])])))^2) = (Subscript[x, 1])+2(Sqrt[(Subscript[x, 1])(Subscript[x, 2])])+(Subscript[x, 2]) = 4+2(Sqrt[1]) = 6 => (Sqrt[(Subscript[x, 1])])+(Sqrt[(Subscript[x, 2])]) = (Sqrt[6]) = (6^(1/2));
((((Sqrt[(Subscript[x, 1])])+(Sqrt[(Subscript[x, 2])])))^(-4)) = ((((6^(1/2))))^(-4)) = (6^(-2)) = (1/(6^2)) = (1/36);
N[(1/36)] = ?;
*)
N[(1/36)]

(* Задача 2. *)
RegionPlot[4(y^2)-3(x^2)≥8&&(x^3)+(y^2)≥1,{x,-2,3},{y,-3,3}]

(* Задача 3. *)
Clear[f];
f[x_]:=(x^5)-5(x^3)+4x;
g[x_]:=4(x^4)+3;
(* частно *)
PolynomialQuotient[f[x],g[x],x]

(* остатък *)
PolynomialRemainder[f[x],g[x],x]

h[x_]:=(1/(f[x]));
Apart[h[x]]

(∑_(n=3)^∞)(1/(f[n]))

(* Задача 4. *)
A=Table[Table[If[i==j,-2,If[i+1==j||i==j+1,1,0]],{j,1,10}],{i,1,10}];
MatrixForm[A]

Det[A]

MatrixForm[Inverse[A]]