Quadratic Equation

Author

Eric W. Weisstein
January 24, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/QuadraticEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/QuadraticEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Series Reversion

s=InverseSeries[Series[a (x^2)+b x+c,{x,0,5}],y]

Normal[s]/.y->0

Table[-(((c^(k+1)) (a^k) Binomial[2 k,k])/((b^(2 k+1)) (k+1))),{k,0,4}]

Sum[-(((c^(k+1)) (a^k) Binomial[2 k,k])/((b^(2 k+1)) (k+1))),{k,0,∞}]

Newton's Relations

Simplify[RootReduce[Plus@@((x/.{ToRules[Roots[(Subscript[a, 2 ])x^2+(Subscript[a, 1]) x+(Subscript[a, 0])==0,x]]}))]]

Simplify[RootReduce[Plus@@((x/.{ToRules[Roots[(Subscript[a, 2 ])x^2+(Subscript[a, 1]) x+(Subscript[a, 0])==0,x]]})^2)]]

Simplify[RootReduce[Plus@@((x/.{ToRules[Roots[(Subscript[a, 2 ])x^2+(Subscript[a, 1]) x+(Subscript[a, 0])==0,x]]})^3)]]

Simplify[RootReduce[Plus@@((x/.{ToRules[Roots[(Subscript[a, 2 ])x^2+(Subscript[a, 1]) x+(Subscript[a, 0])==0,x]]})^4)]]

Factorable

FactorableQ[p_,x_]:=Module[{f=Rest[FactorList[p]]},
Length[f]>1||f[[1,2]]>1
]

Solve[{a+b==a1,a b==a0},{a,b}]

[-n,n]

Table[Count[x^2+#1x+#2&@@@Flatten[Outer[List,Range[-n,n],Range[-n,n]],1],
_?(FactorableQ[#,x]&)],{n,0,10}]

With[{n=1},Select[x^2+#1x+#2&@@@Flatten[Outer[List,Range[-n,n],Range[-n,n]],1],
(FactorableQ[#,x]&)]]

Table[Length[Select[Flatten[Outer[List,Range[-n,n],Range[-n,n]],1],IntegerQ[(1/2) (#[[2]]+(Sqrt[-4 #[[1]]+((#[[2]])^2)]))]&]],{n,0,10}]

A067274

a[0]:=1
a[1]:=4
a[n_]:=a[n]=a[n-1]+2(DivisorSigma[0,n]+1)+If[IntegerQ[Sqrt[n]],1,0]

a/@Range[0,10]

[0,n]

Table[Count[x^2+#1x+#2&@@@Flatten[Outer[List,Range[0,n],Range[0,n]],1],
_?(FactorableQ[#,x]&)],{n,0,10}]

b[n_]:=Length[Select[Flatten[Outer[List,Range[0,n],Range[0,n]],1],IntegerQ[(1/2) (#[[2]]+(Sqrt[-4 #[[1]]+((#[[2]])^2)]))]&]]

b/@Range[0,10]

[1,n]

Table[Count[x^2+#1x+#2&@@@Flatten[Outer[List,Range[n],Range[n]],1],
_?(FactorableQ[#,x]&)],{n,0,10}]

#==Factor[#]&/@With[{n=4},Select[x^2+#1x+#2&@@@Flatten[Outer[List,Range[n],Range[n]],1],
(FactorableQ[#,x]&)]]//ColumnForm//TraditionalForm

Factor/@%//Trad

c[n_]:=Length[Select[Flatten[Outer[List,Range[n],Range[n]],1],IntegerQ[(1/2) (#[[2]]+(Sqrt[-4 #[[1]]+((#[[2]])^2)]))]&]]

c/@Range[0,10]

Plots

<<Utilities`Plot`

Show[
Block[{$DisplayFunction=Identity},
{
IntegerListPlot[MapIndexed[#1/(2#2[[1]]+1)^2&,a/@Range[0,50]]],
IntegerListPlot[MapIndexed[#1/(#2[[1]]+1)^2&,b/@Range[0,50]],PlotStyle->Blue],
IntegerListPlot[MapIndexed[#1/(#2[[1]])^2&,c/@Range[0,50]],PlotStyle->Green]
}],
PlotRange->All,AxesLabel->TraditionalForm/@{x,f}
]