Quadratic equation with integral coefficients [closed]

Let $$a,b,c$$ be Natural Numbers, such that roots of the equation $$ax^2+bx+c=0$$ are distinct and both lie in the interval

1. (0,1)
2. (1,2)
3. (2,3)

(Brackets signify open interval, roots are $$IN BETWEEN$$ the numbers in each part.)

Find minimum possible value of $$a, b, c.$$

On my part, I solved for part 1, i.e. for distinct roots between (0,1). But for the next two parts, the things are getting a too bit messy.

While it may have similarity in question for given part 1 in stack exchange, there is no generalized method so that we can solve for other such intervals.

• Welcome to the Mathematica StackExchange. Please reformulate your question as a specific question about Wolfram Mathematica, otherwise it may get closed as off-topic. – Roman May 5 '19 at 11:50
• What wolfram mathematica is going to do about it? How is this question even related to it? Please, I didn't understand, sorry. Do you have any solution to this question, by the way? – user10595795 May 5 '19 at 11:59
• I think it's more about quadratics, but I couldn't find that suitable tag – user10595795 May 5 '19 at 12:00
• Please read here: mathematica.stackexchange.com/help/on-topic – Roman May 5 '19 at 12:15
• Sorry, it's not related to mathematica. Please flag this as off topic. I mistakenly submitted it to mathematica instead of mathematics. I am extremely sorry. – user10595795 May 5 '19 at 12:16

I felt this was an interesting problem to analyze numerically in Mathematica: suppose I don't know how to analyze this problem analytically, how could I do so numerically? This is one way:

First, just plot where the descriminate is positive, then just do a brute search for the solution. Note in the code below there are no solutions in the intervals 1<(a,b,c)<50.

rp1 = RegionPlot3D[
b^2 - 4 a c > 0 , {a, 1, 10000}, {b, 1, 10000},
{c,
1, 10000},
AxesLabel -> {"a", "b", "c"}, PlotStyle ->
{Opacity[0.572]},
Mesh -> None]
myData = {};
myVals = Table[If[b^2 - 4 a c > 0 ,
myVals = x /. NSolve[a x^2 + b x + c == 0];
theResults = (1 < # < 2) & /@ myVals;
If[! MemberQ[theResults, False],
myData = Append[myData, {a, b, c}];
];
];
, {a, 1, 50}, {b, 1, 50}, {c, 1, 50}];
myData
myPoints = Point @@ {myData};
myPlot1 = Show[Graphics3D[myPoints]]
Show[{rp1, myPlot1}]