# Quickest way to solve $x^2+(30-14\mathrm i)x=240$

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 $$x=6+2\mathrm i$$.

• In general, the quickest way is to give it to Mathematica Commented Sep 8, 2023 at 13:40
• What's wrong with quadratic formula? Of course, it's essentially the same as completing the square, but it skips over the manual work. Commented Sep 8, 2023 at 13:45
• Note that $210=2\times 21\times 5$ and $416=21^2-5^2$ so you immediately get $21-5i$ as a square-root. Commented Sep 8, 2023 at 13:48
• 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. Commented Sep 8, 2023 at 13:50
• 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.
– Jam
Commented Sep 8, 2023 at 13:51

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