I want the quickest way to solve for $x$ in $x^2+(30-14\mathrm i)x=240$. Moreover, is there any method that can solve quadratics with complex coefficients quickly?
Here, quick/slow are assessed based on the time needed to work entirely by hand.
The slow method is, \[(x-7\mathrm i+15)^2=416-210\mathrm i.\] Then we calculate the square root of the magnitude and half the argument of $416-210\mathrm i$, which took about $6$ minutes to do by hand. The roots are $x=12\mathrm i-36$ or $x=6+2\mathrm i$.
$$x^2-(-30+14i)x-240=0$$ A good question for students/youth. If the roots are $x_1=a+bi$ and $x_2=c+di$ then by Vieta's formulas $$x_1+x_2=(a+c)+(b+d)i=-30+14i$$ $$x_1x_2=(ac-bd)+(ad+bc)i=-240+0i$$ $$a+c=-30$$ $$b+d=14$$ $$ac-bd=-240$$ $$ad+bc=0$$ From the last equation $c=-ka, d=kb$. Hence, $$(k-1)a=30$$ $$(k+1)b=14$$ $$k(a^2+b^2)=240.$$ By guessing $k=6,a=6,b=2$. Answer: $x_1=6+2i$, $x_2=-36+12i.$