I'm in difficulty in the study of complex numbers. I have this equation $w_1z^2+w_2z^2+i=0$ and the solution of this equation is $z=3+i$ and $z=1/2+i$. I had to find $w_1$ and $w_2$
I know that I must to search the results using this formula \begin{array}…\end{array} $$ \left\{ \begin{array}{c} x_1+x_2= \\ x_1*x_2=\\ \end{array} \right. $$ So for calculate the first $x_1+x_2$ I'm using this formula: (a+ib)+(c+id)=(a+c)+i(b+d), and I see that 1/2+i it's a fraction. So for this I'm going to aplicate this one: $(\frac{a+ib}{c+id})=(\frac{2+i+0-0}{2^2+i^2})=(\frac{2+i}{3})$ \begin{array}…\end{array} $$ \left\{ \begin{array}{c} x_1+x_2= (3+i)+(\frac{1}{2+i})=(\frac{11}{3})+2i\\ x_1*x_2=(3+i)(\frac{2+i}{3+i})=\frac{2+i}{1}=2+i\\ \end{array} \right. $$ And I see the the result it's different from the book, and I can't to find the correct solution. what's wrong?
I've already noted in the comment that you are mis-interpreting "$1/2 +i$". It is also apparent that you have mis-copied the equation. It should either be $$w_1z^2 + w_2z +i = 0$$ or $$w_1z + w_2z^2 +i = 0$$ It is absolutely pointless to square $z$ in both the $w_1$ and $w_2$ terms. Because I prefer the indices to line up with the powers, I'll assume it is the latter version, written in order as $$w_2z^2 + w_1z_1 + i = 0$$
Now, there are different ways to approach this, but the most straightforward is just to note that you are given that the two values $z = 3 + i$ and $z = \frac 12 + i$ both make the equation true. Note that $$(3+i)^2 = 3^2 + 2\cdot 3 \cdot i + i^2 = 8 + 6i\\\left(\frac12 + i\right)^2 = \frac 1{2^2} + 2\cdot \frac 12\cdot i + i^2 = -\frac 34 + i$$ So the first value gives us the equation $$(8 + 6i)w_2 + (3 + i)w_1 + i = 0$$ And the second value gives us the equation $$\left(-\frac 34 + i\right)w_2 + \left(\frac 12 + i\right)w_1 + i = 0$$ Multiplying through by $4$ makes it look a little nicer: $$(-3 + 4i)w_2 + (2 + 4i)w_1 + 4i = 0$$
So now you have a system of two equations in the two unknowns $w_1$ and $w_2$, which you can solve to find the needed values.