Answer:
[tex]a=\displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}i\\\\a=\displaystyle\frac{1}{2}-\displaystyle\frac{\sqrt{3}}{2}i[/tex]
[tex]b=\displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}i\\\\b=\displaystyle\frac{1}{2}-\displaystyle\frac{\sqrt{3}}{2}i[/tex]
Step-by-step explanation:
Recall that
[tex]i=\sqrt{-1}[/tex]
so
[tex]i^2=-1[/tex]
we have then
[tex]a^2+b^2=-1\\\\a+b=1[/tex]
Isolating b in the second equation we get
b = 1-a
Replace this value in the first equation
[tex]a^2+(1-a)^2=-1\Rightarrow a^2+1-2a+a^2=-1\Rightarrow\\\\\Rightarrow 2a^2-2a+2=0\Rightarrow a^2-a+1=0[/tex]
Solving the quadratic equation we get two possible solutions for a
[tex]a=\displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}i\\\\a=\displaystyle\frac{1}{2}-\displaystyle\frac{\sqrt{3}}{2}i[/tex]
Replacing these values in b=1-a, we get two possible solutions for b
[tex]b=\displaystyle\frac{1}{2}+\displaystyle\frac{\sqrt{3}}{2}i\\\\b=\displaystyle\frac{1}{2}-\displaystyle\frac{\sqrt{3}}{2}i[/tex]
