Solve the equation $x^2+bx+c+t\log(x)=0$

Question

There is a explicit formula to solve the equation

$x^2+bx+c+t\log(x)=0$

with the constraint $x>0$?

Answer 1

No. It's a transcendental equation. I'd use

$$e^{-x^2} = x^t e^{c + b x}$$ or completing the square and redefining the constant $c\gets \exp(c-b^2/4)$:

$$c x^t = e^{-(x+b/2)^2} $$

Or Mathematica's FindRoot, etc.