Revisions to How can I choose the integer numbers a, b, c, d, t so that the equation $\sqrt{a x^2+b x+c}=d x+t$ has two integer solutions?

deleted 237 characters in body

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edited Nov 18, 2023 at 22:52

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No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtkra = RandomChoice[Complement[Range[-10,10;
n 10],= 24;

Table[{0}]r1, 3];
r1r2r2} = RandomSample[Range[-10ra, 10]ra], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 2];
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2}= RandomChoice[Complement[Range[->ra, r1r2]

Solve[%]

Clear[dtkra], r1r2]

{-50}], 4}
2];
129 + 108 x +t 60= x^2RandomChoice@Complement[Range[0, ==2 (7ra] + 8 x)^2

{{x -> Max[-5},d {x -> 4}}

Update:

Table[dtk = RandomChoice[Complement[Range[-10r1, 10]r2}], {0}], 3]; 
  r1r2 = RandomSample[Range[-10, 10], ];
    2]; Sqrt[(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
      k r1 r2r2] == (d x + t)^2 /. Thread[{d, tn];

Partition[%, k} -> dtk]2] /./ Grid
 
Clear[d, k, Thread[{r1, r2} -> r1r2], 100];

Grid@Partition[
  Select[Sqrt[#[[1]]] == (#[[2]] /. x__^2 :> x) & /@ %, 
   Length@Solve[#] == 2 &]t, 2]
Clear[dtkra, r1r2]n]

$\begin{array}{ll} \sqrt{75 x^2+84 x-27}=9 x+3 & \sqrt{6 x^2-4 x+103}=7-2 x \\ \sqrt{61 x^2-40 x+1}=1-8 x & \sqrt{42 x^2+24 x+478}=6 x+10 \\ \sqrt{x^2-100 x-316}=-3 x-2 & \sqrt{5 x^2-26 x+148}=10-2 x \\ \sqrt{74 x^2+250 x+745}=-8 x-5 & \sqrt{4 x^2-x-5}=x+1 \\ \sqrt{97 x^2-164 x+28}=7-10 x & \sqrt{9 x^2+64 x-104}=4 x+1 \\ \sqrt{40 x^2-119 x-38}=4-7 x & \sqrt{21 x^2-58 x+25}=5-5 x \\ \sqrt{62 x^2+142 x+72}=-8 x-10 & \sqrt{50 x^2-99 x+86}=7 x-6 \\ \sqrt{79 x^2-38 x-124}=9 x-4 & \sqrt{52 x^2+131 x+79}=-7 x-7 \\ \sqrt{105 x^2+200 x+99}=-10 x-8 & \sqrt{29 x^2+111 x-82}=6 x+4 \\ \sqrt{76 x^2+101 x-176}=9 x+2 & \sqrt{2 x^2-30 x-56}=2-2 x \\ \sqrt{59 x^2-230 x+300}=7 x-10 & \sqrt{83 x^2+64 x+109}=9 x+5 \\ \sqrt{97 x^2-98 x+16}=4-10 x & \sqrt{29 x^2+60 x+32}=5 x+6 \\ \end{array}$$\begin{array}{ll} \sqrt{46 x^2-498 x+1356}=34-6 x & \sqrt{103 x^2-1957 x+9334}=98-10 x \\ \sqrt{7 x^2+108 x+486}=2 x+27 & \sqrt{4 x^2+165 x+920}=3 x+30 \\ \sqrt{63 x^2+1047 x+4348}=8 x+66 & \sqrt{77 x^2-1584 x+8100}=90-9 x \\ \sqrt{27 x^2+576 x+2808}=6 x+54 & \sqrt{21 x^2+188 x+288}=5 x+16 \\ \sqrt{4 x^2+61 x+151}=3 x+11 & \sqrt{23 x^2-112 x+168}=14-4 x \\ \sqrt{103 x^2-1550 x+5848}=76-10 x & \sqrt{28 x^2+340 x+1001}=6 x+31 \\ \sqrt{-6 x^2-85 x-271}=-x-3 & \sqrt{104 x^2-852 x+1753}=43-10 x \\ \sqrt{62 x^2+696 x+1939}=8 x+43 & \sqrt{90 x^2+729 x+1573}=9 x+41 \\ \sqrt{73 x^2+825 x+2331}=8 x+51 & \sqrt{3 x^2+18 x+76}=x+10 \\ \sqrt{30 x^2-370 x+1189}=37-5 x & \sqrt{77 x^2-1954 x+12201}=109-9 x \\ \sqrt{44 x^2+677 x+2404}=7 x+48 & \sqrt{-2 x^2+46 x+496}=2 x+16 \\ \sqrt{35 x^2-690 x+3631}=61-5 x & \sqrt{75 x^2-492 x+793}=9 x-31 \\ \end{array}$

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
r1r2 = RandomSample[Range[-10, 10], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2} -> r1r2]

Solve[%]

Clear[dtk, r1r2]

{-5, 4}

129 + 108 x + 60 x^2 == (7 + 8 x)^2

{{x -> -5}, {x -> 4}}

Update:

Table[dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3]; 
  r1r2 = RandomSample[Range[-10, 10], 
    2]; (d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
      k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
   Thread[{r1, r2} -> r1r2], 100];

Grid@Partition[
  Select[Sqrt[#[[1]]] == (#[[2]] /. x__^2 :> x) & /@ %, 
   Length@Solve[#] == 2 &], 2]
Clear[dtk, r1r2]

$\begin{array}{ll} \sqrt{75 x^2+84 x-27}=9 x+3 & \sqrt{6 x^2-4 x+103}=7-2 x \\ \sqrt{61 x^2-40 x+1}=1-8 x & \sqrt{42 x^2+24 x+478}=6 x+10 \\ \sqrt{x^2-100 x-316}=-3 x-2 & \sqrt{5 x^2-26 x+148}=10-2 x \\ \sqrt{74 x^2+250 x+745}=-8 x-5 & \sqrt{4 x^2-x-5}=x+1 \\ \sqrt{97 x^2-164 x+28}=7-10 x & \sqrt{9 x^2+64 x-104}=4 x+1 \\ \sqrt{40 x^2-119 x-38}=4-7 x & \sqrt{21 x^2-58 x+25}=5-5 x \\ \sqrt{62 x^2+142 x+72}=-8 x-10 & \sqrt{50 x^2-99 x+86}=7 x-6 \\ \sqrt{79 x^2-38 x-124}=9 x-4 & \sqrt{52 x^2+131 x+79}=-7 x-7 \\ \sqrt{105 x^2+200 x+99}=-10 x-8 & \sqrt{29 x^2+111 x-82}=6 x+4 \\ \sqrt{76 x^2+101 x-176}=9 x+2 & \sqrt{2 x^2-30 x-56}=2-2 x \\ \sqrt{59 x^2-230 x+300}=7 x-10 & \sqrt{83 x^2+64 x+109}=9 x+5 \\ \sqrt{97 x^2-98 x+16}=4-10 x & \sqrt{29 x^2+60 x+32}=5 x+6 \\ \end{array}$

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

ra = 10;
n = 24;

Table[{r1, r2} = RandomSample[Range[-ra, ra], 2];
  {d, k} = RandomChoice[Complement[Range[-ra, ra], {0}], 2];
  t = RandomChoice@Complement[Range[0, 2 ra] + Max[-d {r1, r2}], {0}];
  Sqrt[(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
     k r1 r2] == (d x + t), n];

Partition[%, 2] // Grid

Clear[d, k, r1, r2, t, ra, n]

$\begin{array}{ll} \sqrt{46 x^2-498 x+1356}=34-6 x & \sqrt{103 x^2-1957 x+9334}=98-10 x \\ \sqrt{7 x^2+108 x+486}=2 x+27 & \sqrt{4 x^2+165 x+920}=3 x+30 \\ \sqrt{63 x^2+1047 x+4348}=8 x+66 & \sqrt{77 x^2-1584 x+8100}=90-9 x \\ \sqrt{27 x^2+576 x+2808}=6 x+54 & \sqrt{21 x^2+188 x+288}=5 x+16 \\ \sqrt{4 x^2+61 x+151}=3 x+11 & \sqrt{23 x^2-112 x+168}=14-4 x \\ \sqrt{103 x^2-1550 x+5848}=76-10 x & \sqrt{28 x^2+340 x+1001}=6 x+31 \\ \sqrt{-6 x^2-85 x-271}=-x-3 & \sqrt{104 x^2-852 x+1753}=43-10 x \\ \sqrt{62 x^2+696 x+1939}=8 x+43 & \sqrt{90 x^2+729 x+1573}=9 x+41 \\ \sqrt{73 x^2+825 x+2331}=8 x+51 & \sqrt{3 x^2+18 x+76}=x+10 \\ \sqrt{30 x^2-370 x+1189}=37-5 x & \sqrt{77 x^2-1954 x+12201}=109-9 x \\ \sqrt{44 x^2+677 x+2404}=7 x+48 & \sqrt{-2 x^2+46 x+496}=2 x+16 \\ \sqrt{35 x^2-690 x+3631}=61-5 x & \sqrt{75 x^2-492 x+793}=9 x-31 \\ \end{array}$

added 1250 characters in body

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edited Nov 18, 2023 at 20:41

azerbajdzan

25.1k
2
22
62

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
r1r2 = RandomSample[Range[-10, 10], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2} -> r1r2]

Solve[%]

Clear[dtk, r1r2]

{-5, 4}

129 + 108 x + 60 x^2 == (7 + 8 x)^2

{{x -> -5}, {x -> 4}}

Update:

Table[dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3]; 
  r1r2 = RandomSample[Range[-10, 10], 
    2]; (d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
      k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
   Thread[{r1, r2} -> r1r2], 100];

Grid@Partition[
  Select[Sqrt[#[[1]]] == (#[[2]] /. x__^2 :> x) & /@ %, 
   Length@Solve[#] == 2 &], 2]
Clear[dtk, r1r2]

$\begin{array}{ll} \sqrt{75 x^2+84 x-27}=9 x+3 & \sqrt{6 x^2-4 x+103}=7-2 x \\ \sqrt{61 x^2-40 x+1}=1-8 x & \sqrt{42 x^2+24 x+478}=6 x+10 \\ \sqrt{x^2-100 x-316}=-3 x-2 & \sqrt{5 x^2-26 x+148}=10-2 x \\ \sqrt{74 x^2+250 x+745}=-8 x-5 & \sqrt{4 x^2-x-5}=x+1 \\ \sqrt{97 x^2-164 x+28}=7-10 x & \sqrt{9 x^2+64 x-104}=4 x+1 \\ \sqrt{40 x^2-119 x-38}=4-7 x & \sqrt{21 x^2-58 x+25}=5-5 x \\ \sqrt{62 x^2+142 x+72}=-8 x-10 & \sqrt{50 x^2-99 x+86}=7 x-6 \\ \sqrt{79 x^2-38 x-124}=9 x-4 & \sqrt{52 x^2+131 x+79}=-7 x-7 \\ \sqrt{105 x^2+200 x+99}=-10 x-8 & \sqrt{29 x^2+111 x-82}=6 x+4 \\ \sqrt{76 x^2+101 x-176}=9 x+2 & \sqrt{2 x^2-30 x-56}=2-2 x \\ \sqrt{59 x^2-230 x+300}=7 x-10 & \sqrt{83 x^2+64 x+109}=9 x+5 \\ \sqrt{97 x^2-98 x+16}=4-10 x & \sqrt{29 x^2+60 x+32}=5 x+6 \\ \end{array}$

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
r1r2 = RandomSample[Range[-10, 10], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2} -> r1r2]

Solve[%]

Clear[dtk, r1r2]

{-5, 4}

129 + 108 x + 60 x^2 == (7 + 8 x)^2

{{x -> -5}, {x -> 4}}

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
r1r2 = RandomSample[Range[-10, 10], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2} -> r1r2]

Solve[%]

Clear[dtk, r1r2]

{-5, 4}

129 + 108 x + 60 x^2 == (7 + 8 x)^2

{{x -> -5}, {x -> 4}}

Update:

Table[dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3]; 
  r1r2 = RandomSample[Range[-10, 10], 
    2]; (d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
      k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
   Thread[{r1, r2} -> r1r2], 100];

Grid@Partition[
  Select[Sqrt[#[[1]]] == (#[[2]] /. x__^2 :> x) & /@ %, 
   Length@Solve[#] == 2 &], 2]
Clear[dtk, r1r2]

$\begin{array}{ll} \sqrt{75 x^2+84 x-27}=9 x+3 & \sqrt{6 x^2-4 x+103}=7-2 x \\ \sqrt{61 x^2-40 x+1}=1-8 x & \sqrt{42 x^2+24 x+478}=6 x+10 \\ \sqrt{x^2-100 x-316}=-3 x-2 & \sqrt{5 x^2-26 x+148}=10-2 x \\ \sqrt{74 x^2+250 x+745}=-8 x-5 & \sqrt{4 x^2-x-5}=x+1 \\ \sqrt{97 x^2-164 x+28}=7-10 x & \sqrt{9 x^2+64 x-104}=4 x+1 \\ \sqrt{40 x^2-119 x-38}=4-7 x & \sqrt{21 x^2-58 x+25}=5-5 x \\ \sqrt{62 x^2+142 x+72}=-8 x-10 & \sqrt{50 x^2-99 x+86}=7 x-6 \\ \sqrt{79 x^2-38 x-124}=9 x-4 & \sqrt{52 x^2+131 x+79}=-7 x-7 \\ \sqrt{105 x^2+200 x+99}=-10 x-8 & \sqrt{29 x^2+111 x-82}=6 x+4 \\ \sqrt{76 x^2+101 x-176}=9 x+2 & \sqrt{2 x^2-30 x-56}=2-2 x \\ \sqrt{59 x^2-230 x+300}=7 x-10 & \sqrt{83 x^2+64 x+109}=9 x+5 \\ \sqrt{97 x^2-98 x+16}=4-10 x & \sqrt{29 x^2+60 x+32}=5 x+6 \\ \end{array}$

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created Nov 18, 2023 at 14:22

azerbajdzan

25.1k
2
22
62

No need to use FindInstance. In fact you can generate all possible equations with prescribed roots r1 and r2.

dtk = RandomChoice[Complement[Range[-10, 10], {0}], 3];
r1r2 = RandomSample[Range[-10, 10], 2]

(d^2 - k) x^2 + (k (r1 + r2) + 2 d t) x + t^2 - 
    k r1 r2 == (d x + t)^2 /. Thread[{d, t, k} -> dtk] /. 
 Thread[{r1, r2} -> r1r2]

Solve[%]

Clear[dtk, r1r2]

{-5, 4}

129 + 108 x + 60 x^2 == (7 + 8 x)^2

{{x -> -5}, {x -> 4}}

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