Solve the equation $z^3-2z^2+3z-2=0.$

Question

Solve the following equation.

$$z^3-2z^2+3z-2=0$$

If $a$ is a complex solution of this equation, what does $A$ equal?

$$A= \frac{|a|^2}{1-i ^ {43}}$$

It's on my exams and I really need to solve this on to pass... Any help?

Answer 1

Since $|z_{1}|=|z_{2}|=\sqrt{(\frac{1}{2})^{2}+(\frac{\pm\sqrt{7}}{2})^{2}}=\sqrt{2}$ and $i^{43}=(i^{2})^{21}i=-i$ we have $$A=\frac{|a|^{2}}{1+i}=\frac{2}{1+i}=\frac{2}{1+i}\times\frac{1-i}{1-i}=\frac{2(1-i)}{2}=1-i.$$

Answer 2

Hint: $z^{3}-2z^{2}+3z-2=(z-1)(z^{2}-z+2)$. You get the complex (I mean non-real) roots by solving then quadratic $z^{2}-z+2=0$.