First slide

Algebra of complex numbers

Question

If $8 i z^{3} + 12 z^{2} - 18 z + 27 i = 0$

Moderate

A

$| z | = \frac{3}{2}$
B

$|z| = \frac{2}{3}$
C

$|z| = 1$
D

$|z| = \frac{3}{4}$

Solution

$\begin{array}{l} 8 i z^{3} + 12 z^{2} - 18 z + 27 i = 0 \\ \Rightarrow 8 i z^{3} - 12 i^{2} z^{2} - 18 z + 27 i = 0 \\ \Rightarrow 4 i z^{2} (2 z - 3 i) - 9 (2 z - 3 i) = 0 \end{array}$
$\begin{array}{lr} \Rightarrow (4 i z^{2} - 9) (2 z - 3 i) = 0 \\ \Rightarrow z^{2} = \frac{9}{4 i} or z = \frac{3 i}{2} \end{array}$
In any case $| z | = 3 / 2$

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