"produce only identical expression with "e1", not with "e2" as expected?!"
What makes you think e2 is the expected output?
There is no unique simplified expression for most of the expressions, (one such exception being - sin^2 + cos^ == 1, for any input). This is mentioned in the documentation of simplify as well - "There is no universal idea as to which form of an expression is simplest."
Let's see how many simplified expressions we can obtain with the given expression -
syms a b c
e1 = -(b*(b^2 - 4*a*c)^(1/2) - 4*a*c + b^2)/(2*a*(b^2 - 4*a*c)^(1/2));
e2 = -(b + (b^2 - 4*a*c)^(1/2))/(2*a);
out1=simplify(e1,'Steps',50,'All',true)
8 outputs for 50 steps. The # of outputs is 10 for 500 steps (After 500, the # of outputs doesn't seem to change regardless of # of steps, I tried till 5e6)
Now, which one should be "the" simplified expression?
"Is there any possibility to use "simplify" command in a different way, to get expected result?"
Unfortunately no, it doesn't look like you will obtain the solution in the form you want. The closest option might be the 3rd element of out.
