Pensieve header: Solving the quadratic equation.

MakeExpression[RadicalBox[RowBox[{p_,":",a_}],n_], StandardForm] := HoldComplete[PersistentRoot[ToExpression@p,ToExpression@n,ToExpression@a]];
PersistentRoot[p_, n_, a_] := PersistentRoot[p,n] = If[NumberQ[PersistentRoot[p,n]],
MinimalBy[
N[(a^(1/n))]Table[(E^(2π I k/n)), {k,n}],
Abs[#-PersistentRoot[p,n]]&
][[1]],
(a^(1/n))
];

((test0:8)^(1/3))

b1={{1,1}/2};
{LocatorPane[Dynamic[b1],Graphics[{Pink,Rectangle[{-1,-1},{1,1}]}]], Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black, Point[r=((test2:b1[[1,1]]+I b1[[1,2]])^(1/2)); {Re[r], Im[r]}]}, PlotRange->{-1,1}]]}

{λ1,λ2}={{1,1}/2, {1,-1}/2};
GraphicsRow[{
LocatorPane[Dynamic[{λ1,λ2}],Graphics[{Pink,Rectangle[{-1,-1},{1,1}]}]], Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black,
{c,b,a}=CoefficientList[(x-{1,I}.λ1)(x-{1,I}.λ2), x];
Text["a", {Re[a],Im[a]}], Text["b", {Re[b],Im[b]}], Text["c",{Re[c],Im[c]}]
}, PlotRange->{-1,1}]],
Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black,
d=(b^2)-4 a c;
Text["d",{Re[d], Im[d]}]
}, PlotRange->{-1,1}]],
Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black,
dr=((disc:d)^(1/2));
Text["(Sqrt[d])", {Re[dr], Im[dr]}]
}, PlotRange->{-1,1}]],
Dynamic[Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black,
r=((-b+((disc:(b^2)-4 a c)^(1/2)))/(2a));
Text["r", {Re[r], Im[r]}]
}, PlotRange->{-1,1}]]
}]

Manipulate[{c,b,a}=CoefficientList[(x-{1,I}.λ1)(x-{1,I}.λ2), x];
Graphics[{LightBlue, Rectangle[{-1,-1},{1,1}],Black,
r=((-b+((disc:(b^2)-4 a c)^(1/2)))/(2a));
Point[{Re[r], Im[r]}]
}, PlotRange->{-1,1}],
{{λ1,{1,1}/2,"λ1"},{-1,-1},{1,1}},
{{λ2,{1,-1}/2,"λ2"},{-1,-1},{1,1}}, ControlPlacement->Left]