If not, please do correct me.

The question is,

The complex number z is given by z = 1 - i. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$.

I don't know, I just feel that something is not right here even though I've checked many times. Here's my answer:

$\displaystyle -1/2 - (5/2)i$

Ok. I've figured out how to use LaTex.

The complex number z is given by $\displaystyle z = 1 - i$. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$

Here is how I expressed it in the form p + qi:

=$\displaystyle z^2 - \frac{1}{z}$

=$\displaystyle (1 - i)^2 - \frac{1}{1 - i}$

=$\displaystyle (1 - 2i - 1) - \frac{1}{1 - i}$

=$\displaystyle \frac{(1 - 2i - 1)(1 - i) - 1}{1 - i}$

=$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}$

=$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}\;\text{x}\,\frac{1 + i}{1 + i}$

=$\displaystyle \frac{(-3 - 2i)(1 + i)}{2}$

=$\displaystyle \frac{-3 - 5i + 2}{2}$

=$\displaystyle \frac{-1 - 5i}{2}$

=$\displaystyle \text{-}\,\frac{1}{2}\;\text{-}\,\frac{5}{2}\;\text{i}$

=Correct or Wrong? Please help to evaluate! Thanks. If this is correct then that means my modulus and argument are also correct.