Question

Nonreal complex numbers z satisfying the equation $z^3+2z^2+3z+2=0$ are

• A. $\frac{-1\pm \sqrt{-7}}{2}$
• B. $\frac{1+\sqrt{7}i}{2},\frac{1-\sqrt{7}i}{2}$
• C. $-i,\frac{-1\pm \sqrt{7}i}{2},\frac{-1-\sqrt{7}i}{2}$
• D. None of these

Answer 1

The correct option is A $\frac{-1\pm \sqrt{-7}}{2}$

$z^3+2z^2+3z+2=0$ ...(1)
$\Rightarrow z{(z+1)}^2+2(z+1)=0$
$\Rightarrow (z+1)(z^2+z+2)=0$
$\Rightarrow z=-1$
$\&z^2+z+2=0$
$\Rightarrow z=\frac{-1\pm \sqrt{1-8}}{2}=\frac{-1\pm \sqrt{-7}}{2}$
Non real roots are $\frac{-1\pm \sqrt{-7}}{2}$