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Expert: - 5/10/2009


Question
Using the quadratric formula or complete the square to find all solutions for z in the equation: z to the second power - (1+i)z+ 1/2+ 1/4i=0

Answer
I don't see what the quadratic equation has to do with this problem.

Move everything that is not z to the other side of the equation.

(1+i)z = - 1/2 - i/4.

Divide by (1+i).
z = (-1/2 - i/4)/(1+i).

However, I do see using what's called a number's conjugate.
Since there is an i in the denominator,
multiply the numerator and denominator by the conjugate.
That will be z = -(1-i)(1/2 + i/4)/((1+i)(1-i)).

In the numerator, -(1-i)(1/2 + i/4) = -(1/2 +i/4 -i/2 - i²/4).
Note that i² = -1, so i²/4 = -1/4.  Also  not that i/2 is 2i/4.

So this reduces to -(1/4 -i/4) = (i - 1)/4
when the negative sign is multiplied through and the /4 is taken out.

Now in the denominator, (1+i)(1-i) = 1² - i² = 1 + 1 = 2.

So putting both of these together gives
z = ((i - 1)/4)/2 = ((i-1)/4)/(2/1).  

Invert the bottom and multiply by 1/2 to get
z = (i-1)/8.

Now the problem asks for z², so the answer is
z² = (1-i)²/64.  (1-i)² = (1-i) - i(1-i) = 1-i - i-1 = -2i.

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