Question

If the four roots of the equation $z^4+z^3+2z^2+z+1=0$ form a quadrilateral on the Argand plane, then the area of the quadrilateral is

• A. $\frac{\sqrt{3}+2}{8}$
• B. $\frac{\sqrt{3}+2}{4}$
• C. $\frac{\sqrt{3}+2}{2}$
• D. $\frac{\sqrt{3}+2}{3}$

Answer 1

The correct option is B $\frac{\sqrt{3}+2}{4}$

$z^4+z^3+2z^2+z+1=0$
On dividing the equation by $z^2$,
$z^2+\frac{1}{z^2}+z+\frac{1}{z}+2=0$
$\Rightarrow {(z+\frac{1}{z})}^2+(z+\frac{1}{z})=0$
$\Rightarrow z+\frac{1}{z}=0$ or $z+\frac{1}{z}+1=0$
$\Rightarrow z=\pm i$ or $z=\omega ,{\omega }^2$
where $\omega$ is cube root of unity.

Now, plotting solution on Argand plane,


The quadrilateral is a trapezium.
Thus, required area $=\frac{1}{2}\times (2+\sqrt{3})\times \frac{1}{2}=\frac{\sqrt{3}+2}{4}$