Question

If $z_1$ and $z_2$ are the complex roots of the equation ${(x-3)}^3+1=0$, then $z_1+z_2$ equals to

• A. $6$
• B. $3$
• C. $5$
• D. $7$

Answer 1

The correct option is A $6$

Let the equation

$\begin{array}{c}{\left(x-3\right)}^3+1=0\\ \\ {\left(x-3\right)}^3=-1\\ \\ \left(x-3\right)=\sqrt{-1}\\ \\ \left(x-3\right)=\pm i\\ \\ x-3=\pm i\end{array}$

where $i$ is the imagery number.

Now,

$\begin{array}{c}x-3=ior-i\\ \\ x=3+ior3-i\end{array}$

Since, two roots are $z_1$ and $z_2$

So,

$z_1=3+i$ or $z_2=3-i$

Now,

$z_1+z_2=3+i+3-i=6$

Hence, sum of the roots is $6$.

Thus option $A$ is correct.