Question

How do you find complex/imaginary roots of the cube root of -27 ?

Answer 1

Do you have DeMoivre?

Answer 2

\(\large x^3=-25\)
\(\large x^3+25=0\)
factor it

Answer 3

\(\large x^3=-27\)
\(\large x^3+27=0\)
factor it
\(\large x^3+27=(x+3)(x^2-3x+9)\)
\(\large (x+3)=0 , x=-3 \text{ (real root)}\)
then usinf general formula for quiadratic equation solve

\(\large x^2-3x+9=0\)
\(\Huge \color{green}{x=\frac {-b \pm \sqrt{b^2-4ac}}{2a}}\)

\(\Huge \color{green}{x=\frac {3 \pm \sqrt{9-36}}{2}}\)
\(\Huge \color{green}{x=\frac {3 \pm \sqrt{-25}}{2}}\)
\(\Huge \color{green}{x=\frac {3 \pm 5i}{2}}\)
soo cubic imaginary root are
\(\Huge \color{blue }{x=\frac {3 + 5i}{2}}\) and \(\Huge \color{blue }{x=\frac {3 - 5i}{2}}\)

Answer 4

thanks, makes a lot of sense

Answer 5

ur wlc :)