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Quadratic Quandary (Posted on 2013-11-04) Difficulty: 3 of 5
g(x) is a quadratic function given by:
g(x) = x2 + 12x + 30.

Determine all possible real roots of this equation:
g(g(g(g(g(x))))) = 0

No Solution Yet Submitted by K Sengupta    
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Some Thoughts roots | Comment 3 of 6 |

All 32 roots I come up with are complex:

 10      gosub *Solve(1,12,30)
 20
 30      end
 40
 50      *Solve(A,B,C)
 60        local Discr,S1,S2
 70        Lvl=Lvl+1
 80        Discr=B*B-4*A*C
 90     '  print Lvl,A;B;C,Discr *** commented out
100           Vr=sqrt(Discr)
110           S1=(-B+Vr)/(2*A)
120           S2=(-B-Vr)/(2*A)
130           if Lvl=5 then
140            :print S1:print S2:Ct=Ct+1:if Ct=20 then stop:endif
150          :else
160            :gosub *Solve(A,B,C-S1)
170            :gosub *Solve(A,B,C-S2)
180          :endif
190        Lvl=Lvl-1
200      return

finds only complex roots:

-5.4337966242328360727+3.7493538961007307981#i
-6.5662033757671639272-3.7493538961007307981#i
-6.188973078708025133+3.7774983525195429567#i
-5.8110269212919748669-3.7774983525195429567#i
-5.8110269212919748669+3.7774983525195429567#i
-6.188973078708025133-3.7774983525195429567#i
-6.5662033757671639272+3.7493538961007307981#i
-5.4337966242328360727-3.7493538961007307981#i
-5.3240048974464779996+4.1515168142615801486#i
-6.6759951025535220003-4.1515168142615801486#i
-6.1303046988704467333+4.1797184724222352339#i
-5.8696953011295532666-4.1797184724222352339#i
-5.8696953011295532666+4.1797184724222352339#i
-6.1303046988704467333-4.1797184724222352339#i
-6.6759951025535220003+4.1515168142615801486#i
-5.3240048974464779996-4.1515168142615801486#i
-5.0824381722026647137+4.5205183160058492112#i
-6.9175618277973352862-4.5205183160058492112#i
-5.9262028894512874294+4.557972442820659469#i
-6.0737971105487125704-4.557972442820659469#i
-5.7285077637707018706+4.5790350095257359616#i
-6.2714922362292981293-4.5790350095257359616#i
-6.5685969932212068322+4.5677757184654518396#i
-5.4314030067787931676-4.5677757184654518396#i
-5.4314030067787931676+4.5677757184654518396#i
-6.5685969932212068322-4.5677757184654518396#i
-6.2714922362292981293+4.5790350095257359616#i
-5.7285077637707018706-4.5790350095257359616#i
-6.0737971105487125704+4.557972442820659469#i
-5.9262028894512874294-4.557972442820659469#i
-6.9175618277973352862+4.5205183160058492112#i
-5.0824381722026647137-4.5205183160058492112#i

The program solves the quadratic, and then adjusts the constant term so that zeros of the new quadratic give each of the respective solutions to the level of quadratic above it.

leading me to think maybe I did something wrong.

However, I tried it out for g(g(x))=0, just two levels of depth:

   10      gosub *Solve(1,12,30)
   20
   30      end
   40
   50      *Solve(A,B,C)
   60        local Discr,S1,S2
   70        Lvl=Lvl+1
   80        Discr=B*B-4*A*C
   90     '  print Lvl,A;B;C,Discr
  100           Vr=sqrt(Discr)
  110           S1=(-B+Vr)/(2*A)
  120           S2=(-B-Vr)/(2*A)
  130           if Lvl=2 then
  140            :print S1:print S2:Ct=Ct+1:if Ct=20 then stop:endif
  150          :else
  160            :gosub *Solve(A,B,C-S1)
  170            :gosub *Solve(A,B,C-S2)
  180          :endif
  190        Lvl=Lvl-1
  200      return
 
and it found two real and two complex roots of g(g(x))=0:
 
-4.4349154199267126834
-7.5650845800732873165
-6.0+1.5650845800732873165#i
-6.0-1.5650845800732873165#i

and the real roots check out:

x=-4.4349154199267126834
OK
g=x*x+12*x+30
OK
?g
-3.5505102572168219016
OK
g2=g*g+12*g+30
OK
?g2
 0.0000000000000000009
OK
x=-7.565084580073287316
OK
g=x*x+12*x+30
OK
?g
-3.5505102572168219021
OK
g2=g*g+12*g+30
OK
?g2
-0.0000000000000000017

So it seems to be doing things right, and I'd have to say g(g(g(g(g(x)))))=0 has no real roots.

However, if the original function had been

g(x) = x^2 + 12x - 30

(note the minus sign)

there would be a full 32 real solutions:

  10      gosub *Solve(1,12,-30)
  20
  30      end
  40
  50      *Solve(A,B,C)
  60        local Discr,S1,S2
  70        Lvl=Lvl+1
  80        Discr=B*B-4*A*C
  90     '  print Lvl,A;B;C,Discr
 100           Vr=sqrt(Discr)
 110           S1=(-B+Vr)/(2*A)
 120           S2=(-B-Vr)/(2*A)
 130           if Lvl=5 then
 140            :print S1:print S2:Ct=Ct+1:if Ct @ 20=0 then stop:endif
 150          :else
 160            :gosub *Solve(A,B,C-S1)
 170            :gosub *Solve(A,B,C-S2)
 180          :endif
 190        Lvl=Lvl-1
 200      return
 
 producing
 
  2.677399691665285459
 -14.677399691665285459
  1.6312882598378428108
 -13.6312882598378428108
  1.5809745966989474563
 -13.5809745966989474563
  0.5196676985597539179
 -12.5196676985597539179
  1.5259742917976827729
 -13.5259742917976827729
  0.4637148358327976046
 -12.4637148358327976046
  0.3950793798219034847
 -12.3950793798219034847
 -0.6929952144349964636
 -11.3070047855650035362
  1.4657314834117522766
 -13.4657314834117522766
  0.4024029320764603025
 -12.4024029320764603025
  0.3325451712275664461
 -12.3325451712275664461
 -0.7574056280321333741
 -11.2425943719678666258
  0.2552881821114838044
 -12.2552881821114838044
 -0.837065006880782983
 -11.1629349931192170169
 -0.9405459485938334381
 -11.0594540514061665618
 -2.0870400591764160871
 -9.9129599408235839128
 
 or for g(x) = x^2 - 12x + 30, a different set of 32 real roots:
 
  12.8633893949020836151
 -0.8633893949020836151
  11.9366969009730132938
  0.0633030990269867061
  12.0298218903769117385
 -0.0298218903769117385
  11.1277399156478250752
  0.8722600843521749247
  12.1561696742637786903
 -0.1561696742637786903
  11.2496198741050014649
  0.750380125894998535
  11.3737396376610935879
  0.626260362338906412
  10.5009947362442789044
  1.4990052637557210955
  12.3513552190559084179
 -0.3513552190559084179
  11.4384840842875927878
  0.561515915712407212
  11.5526152716569331958
  0.4473847283430668041
  10.6706790040363506325
  1.3293209959636493674
  11.7165864386738978124
  0.2834135613261021875
  10.8270756307408749844
  1.1729243692591250155
  10.9793771085997968736
  1.0206228914002031263
  10.1311336270934386736
  1.8688663729065613263


  Posted by Charlie on 2013-11-04 15:53:00
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