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Hi I need to solve an inequality for multiple variables, and I do not understand in a good manner the results. The inequality that I tray to obtain is

$$ a+b+(2*c)+\sqrt{(-2\,c-b-a)^2-4\,(2\,c\,b+b\,a-3\,|d|^2)}\ge 0 $$

where $a$, $b$, $c\in \mathbb{R}$. I try to solve the inequality using

Reduce[{a+b+(2*c)+Sqrt[(-2*c-b-a)^2-4\,(2\,c\,b+b\,a-3*|d|^2)] >= 0 
         && a + b == 1 && c <= a && 0 <= a <= 1}, {d}]

I want to find a solution for d and I am not sure if I can do it only specifying d in the variables space or I need to use the complete list {a,b,c,d}. I know this is a simple question but I am really confused. Thanks in advance.

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  • $\begingroup$ Table[With[{d=RandomReal[{-1000,1000}]+I*RandomReal[{-1000,1000}]}, Simplify[a+b+2*c+ Sqrt[(-2*c-b-a)^2-4(2*c*b+b*a-3*Abs[d]^2)]>=0, a+b==1 && c<=a && 0<=a<=1 && Element[a|b|c,Reals]]],{1000}] $\endgroup$ – Bill May 13 at 1:45
  • $\begingroup$ d=r+I*i; Reduce[a+b+2*c+ Sqrt[(-2*c-b-a)^2-4(2*c*b+b*a-3*Abs[d]^2)]<0 && a+b==1 && c<=a && 0<=a<=1 && Element[a|b|c|r|i,Reals],{r,i}] $\endgroup$ – Bill May 13 at 18:54
  • $\begingroup$ Thanks. So I can only specify the variable that I want to obtain the inequality and not the whole of variables, did I understand correctly? $\endgroup$ – mors May 15 at 0:42
  • $\begingroup$ You can ask for the whole of variables or only some variables. Sometimes Solve or Reduce is faster or slower if you only ask for some variables and I do not know why. Often Reduce or Solve is much slower if you ask it for only Real solutions and I do not know why. But it appears that your original inequality is true for all values of d. $\endgroup$ – Bill May 15 at 4:00
  • $\begingroup$ @Bill Oh, Thanks for the explanation, now I understand more the output of Reduce, $\endgroup$ – mors May 15 at 7:34

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