We apply the Quadratic Formula and get,
#x=[(5-i)+-sqrt{(5-i)^2-4(18+i)}]/2, i.e., #
#x=[(5-i)+-{(25-10i-1)-72-4i}]/2, or, #
#x={(5-i)+-sqrt(-48-14i)}/2,#
#:. x={(5-i)+-isqrt(48+14i)}/2..........................(star).#
So, to find #x,# we need to find #sqrt(48+14i).#
Let, #u+iv=sqrt(48+14i); u,v in RR.#
#:. (u+iv)^2=u^2+2iuv-v^2=48+14i.#
Comparing the Real & Imaginary Parts, we have,
#u^2-v^2=48, and, uv=7.#
Now, #(u^2+v^2)^2=(u^2-v^2)^2+4u^2v^2=48^2+14^2=50^2,#
#:. u^2+v^2=50...(1), and, u^2-v^2=48...(2).#
#(1)+(2), &, (1)-(2)" give, "u=7, v=1.#
#:. sqrt(48+14i)=7+i.#
Finally, from #(star),# we get,
#x={(5-i)pmi(7+i)}/2, i.e., #
#x=2+3i, or, x=3-4i,# are the desired roots!
Enjoy Maths.!
