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+(5)/(x)=0`

Simplify the equation,

`\begin{gathered} x+1+(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=(-1\pm√(1^2-4\cdot\:1\cdot\:5))/(2\cdot\:1) \\ x=(-1\pm√(19)i)/(2\cdot\:1) \\ x=(-1+√(19)i)/(2),\: x_{}=(-1-√(19)i)/(2) \\ x=-(1)/(2)+i(√(19))/(2),\: x=-(1)/(2)-i(√(19))/(2) \end{gathered}`

Answer: The root of equation is,

`x=-(1)/(2)+i\frac{\sqrt[]{19}}{2},\: x=-(1)/(2)-i\frac{\sqrt[]{19}}{2}`