Question

Find all solutions of the equation x+1+5/x=0 and give your answer as a list of complex numbers, such as 3-4i, 5+i.

Answer 1

Given:

$x+1+\frac{5}{x}=0$

Simplify the equation,

$\begin{gathered} x+1+\frac{5}{x}=0 \\ x^2+x+5=0 \\ \text{Compare it with ax}^2+bx+c=0 \\ a=1,b=1,c=5 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-1\pm\sqrt{1^2-4\cdot\:1\cdot\:5}}{2\cdot\:1} \\ x=\frac{-1\pm\sqrt{19}i}{2\cdot\:1} \\ x=\frac{-1+\sqrt{19}i}{2},\: x_{}=\frac{-1-\sqrt{19}i}{2} \\ x=-\frac{1}{2}+i\frac{\sqrt{19}}{2},\: x=-\frac{1}{2}-i\frac{\sqrt{19}}{2} \end{gathered}$

Answer: The root of equation is,

$x=-\frac{1}{2}+i\frac{\sqrt[]{19}}{2},\: x=-\frac{1}{2}-i\frac{\sqrt[]{19}}{2}$