ax2+bx+c=0
{x−12a(−b−√b2−4ac)}{x−12a(−b+√b2−4ac)}=0
x=12a(−b±√b2−4ac)=−b2a±√(b2a)2−ca --- (1)
When,
b2−4ac<0
{x−12a(−b−i√4ac−b2)}{x−12a(−b+i√4ac−b2)}=0
x=−b2a±i√4ac−b22a=−b2a±i√ca−(b2a)2 --- (2)
Consider,
f(x)=ax2+bx+c
For the extrememum,
f′(x)=0
2ax+b=0
x=−b2a
which is the real part of the roots of f(x)=0.
At which the function, f(x) attains the value,
f(x)=ax2+bx+c=a(b2a)2−b22a+c
f(x)=a{ca−(b2a)2}
But from (2), given a≠0,
x=−b2a±i√ca−(b2a)2=−b2a±i√f(x)a
When a is a constant, a given quadratic f(x), √|f(x)a| attains extrememum as does f(x). So the complex part of the roots of f(x)=0 attains extrememum as f(x) attains extrememum. And we have,
where B minimum is at the intersection of real(re) and complex(re).
Furthermore, from (1), at the extrememum (real roots only) for a given quadratic f(x),
x=−b2a±√(b2a)2−ca0
(b2a)2−ca=0
b2−4ac=0
This is why the plot of real(re) also passes through the point where dredT→∞ in the graph above. In general, given,
x=−b2a±√(b2a)2−ca
where a≠0,
dxd{a,b,c}=(−b2a)′{a,b,c}±12√(b2a)2−ca(√(b2a)2−ca)′{a,b,c}
when,
(b2a)2−ca=0
b2−4ac=0
we have,
dxd{a,b,c}→∞
ie. a plot of all real roots x of the quadratic f(x)=0, has a pole when,
b2−4ac=0
and
(√(b2a)2−ca)′{a,b,c}≠0
This pole must exist for a general plot of x in order that the graph curves backwards to give two real roots and at the pole, the plot gives a double root. For this reason,
(√(b2a)2−ca)′{a,b,c}≠0
when
b2−4ac=0
is always true.
Given a material, the condition to narrow its bandgap (if it is an insulator or semiconductor) or to maximize its electric conductivity (if it is a conductor) is the same,
1−ATq2π(mec2)2εo=0
where,
A=τo2qvT{r2T+r2or}3/2.ror
τo is the analogue to μo. Next stop finding τo...even if vT=c, rT, the orbital radius of the positive temperature particle that generates the weak field is still not available.
