simplifying equations [closed]

Question

I have a question that seems easy but I can't solve it. I have an equation like:

eqgama = -8.33208*10^56 g1 + 8.18264*10^56 g1^2 == -1.42093*10^56

I use the following statements to find its discriminant:

poly = (eqgama[[1]] - eqgama[[2]]) // Simplify;
Discriminant[poly, g1];

The answer is: 2.29158*10^113

In another program mathematica itself simplifies eqgama to the following equation:

-1.97163*10^54 g1 + 1.93627*10^54 g1 ^2 == -3.36235*10^53

and so the discriminant becomes: 1.28315*10^108.

I have a loop and I want to plot discriminants versus a definite parameter, but in this loop sometimes Mathematica simplifies equations and sometimes not. So the plot fluctuates. What should I do?

Answer 1

Discriminants are not scale free.

If you do

Discriminant[100 (x - a) (x - b), x] - Discriminant[(x - a) (x - b), x] // FullSimplify

you get

9999 (a - b)^2

So, what you can do is to transform your equations into monomials. Trying to respect your notation (which is not the better):

eqgama[1] = -8.33208*10^56 g1 + 8.18264*10^56 g1^2 == -1.42093*10^56;
eqgama[2] = -1.97163*10^54 g1 + 1.93627*10^54 g1^2 == -3.36235*10^53;

c2[x_] := Coefficient[x, g1, 2]
disc[x_] := Discriminant[#/c2[#], g1] &@(x[[1]] - x[[2]])

then:

disc /@ eqgama /@ {1,2}
(*
{0.342253, 1.03686}
*)

Edit

For plotting such things with a parameter, you could do something like:

eqgama[t_] := Cos@t  g1 + Tan@t g1^2 == Gamma@t; (* an example*)

Plot[disc@eqgama@t, {t, 0, 3}]

Edit

Also, please remember that Mathematica has many powerful tools to study parametric equations behavior.

Following with our toy example:

h[t_, g1_] := Cos@t g1 + Tan@t g1^2 - Gamma@t;
Grid[{Through[{Plot3D,DensityPlot,ContourPlot}[h[t,g], {t,0,Pi}, {g,0,5}]]}, Frame -> All]