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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 $x=6+2\mathrm i$.

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    $\begingroup$ In general, the quickest way is to give it to Mathematica $\endgroup$
    – David Raveh
    Commented Sep 8, 2023 at 13:40
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    $\begingroup$ What's wrong with quadratic formula? Of course, it's essentially the same as completing the square, but it skips over the manual work. $\endgroup$
    – Ennar
    Commented Sep 8, 2023 at 13:45
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    $\begingroup$ Note that $210=2\times 21\times 5$ and $416=21^2-5^2$ so you immediately get $21-5i$ as a square-root. $\endgroup$
    – user10354138
    Commented Sep 8, 2023 at 13:48
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    $\begingroup$ To expand on what user10354138 said, you are looking for solutions of $(a+bi)^2 = 416 -210i$, for real $a,b$, which is equivalent to solving $a^2-b^2 = 416,\ 2ab = -210$. Of course, finding one solution gives you the other immediately, since you take the additive inverse. $\endgroup$
    – Ennar
    Commented Sep 8, 2023 at 13:50
  • $\begingroup$ This is no different from factoring an ordinary quadratic, except that now you have the additional arithmetic of $i$. Use the quadratic formula, complete the square, or find by inspection numbers satisfying $r+s=b/a$ and $rs=c/a$, and so on. Practice factoring and mental arithmetic and you can get quicker. See this question. $\endgroup$
    – Jam
    Commented Sep 8, 2023 at 13:51

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$$x^2-(-30+14i)x-240=0$$ 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: $x_1=6+2i$, $x_2=-36+12i.$

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