Is there a simple way to find the integer part (floor) of the positive root of a quadratic equation?

Question

In order to show the positive root of a quadratic equation in Simple Continued Fractions I map the quadratic equation like;

$cx^2+(d−a)x−b=0\implies x=\frac{ax+b}{cx+d}$

where $a > c$ or perhaps $a>d$.

I believe this tells me that the positive root is somewhere between $\frac{a}{c}$ and $\frac{b}{d}$. I can assign it's average to $x$ to check if the floor of what i get is equal to $x$ or not. If not i can iterate over with the obtained result until their integer parts are equal but perhaps there is a better way.

On the other hand, I can of course go backwards to apply quadratic formula and floor it however it beats the purpose. I just want to know if there might be a shortcut from $x=\frac{ax+b}{cx+d}$

I tried $⌊\frac{a}{c}−(\frac{d}{c}−\frac{b}{a})⌋$ but i fails in some cases.

Answer 1

got to make a phone call

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 74 23 -69

  0  form             74          23         -69


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   74 23 -69   delta  -1
1  form   -69 115 28   delta  4
2  form   28 109 -81   delta  -1
3  form   -81 53 56   delta  1
4  form   56 59 -78   delta  -1
5  form   -78 97 37   delta  3
6  form   37 125 -36   delta  -3
7  form   -36 91 88   delta  1
8  form   88 85 -39   delta  -2
9  form   -39 71 102   delta  1
10  form   102 133 -8   delta  -17
11  form   -8 139 51   delta  2
12  form   51 65 -82   delta  -1
13  form   -82 99 34   delta  3
14  form   34 105 -73   delta  -1
15  form   -73 41 66   delta  1
16  form   66 91 -48   delta  -2
17  form   -48 101 56   delta  2
18  form   56 123 -26   delta  -5
19  form   -26 137 21   delta  6
20  form   21 115 -92   delta  -1
21  form   -92 69 44   delta  2
22  form   44 107 -54   delta  -2
23  form   -54 109 42   delta  3
24  form   42 143 -3   delta  -47
25  form   -3 139 136   delta  1
26  form   136 133 -6   delta  -23
27  form   -6 143 21   delta  6
28  form   21 109 -108   delta  -1
29  form   -108 107 22   delta  5
30  form   22 113 -93   delta  -1
31  form   -93 73 42   delta  2
32  form   42 95 -71   delta  -1
33  form   -71 47 66   delta  1
34  form   66 85 -52   delta  -2
35  form   -52 123 28   delta  4
36  form   28 101 -96   delta  -1
37  form   -96 91 33   delta  3
38  form   33 107 -72   delta  -1
39  form   -72 37 68   delta  1
40  form   68 99 -41   delta  -2
41  form   -41 65 102   delta  1
42  form   102 139 -4   delta  -35
43  form   -4 141 67   delta  2
44  form   67 127 -18   delta  -7
45  form   -18 125 74   delta  1
46  form   74 23 -69


  form   74 x^2  + 23 x y  -69 y^2 

minimum was   3rep   x = 2037246618   y = 2476457963 disc 20953 dSqrt 144  M_Ratio  3.786706
Automorph, written on right of Gram matrix:  
-176678757708873898919  -200257625715180256080
-214769047868454187680  -243431299613933984279
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