Question

The area of the triangle whose vertices are the roots $z^3+{iz}^2+2i=0$ is

• A. $2$
• B. $\frac{3}{2}\sqrt{7}$
• C. $\frac{3}{4}\sqrt{7}$
• D. $\sqrt{7}$

Answer 1

The correct option is A $2$

$f\left(z\right)=z^3+i{z}^2+2i=0$

$f\left(i\right)=i^3+i^3+2i=0$

$z^3+i{z}^2+2i=(z-i)(z^2+2iz-2)$

$z^2-2iz-2=0$

$z=\frac{2i\pm \sqrt{-4+8}}{2}$

$z=\frac{2i\pm 2}{2}=1-i,-i-1$

Area = $\frac{1}{2}\times 2\times 2=2$