קיץ 2021 · מועד מיוחד

$w^2 - 4iw - 4 + 2i = 0$

$w_{1,2} = \dfrac{4i \pm \sqrt{(4i)^2 - 4(-4+2i)}}{2}$

$(4i)^2 - 4(-4+2i) = -16 + 16 - 8i = -8i$

$w_{1,2} = \dfrac{4i \pm \sqrt{-8i}}{2}$

$(x+yi)^2 = x^2 - y^2 + 2xyi = -8i$

$\begin{cases} x^2 - y^2 = 0 \\ 2xy = -8 \end{cases}$

$x^2 = \dfrac{16}{x^2} \;\Longrightarrow\; x^4 = 16 \;\Longrightarrow\; x^2 = 4$

$w_{1,2} = \dfrac{4i \pm (2 - 2i)}{2} = 2i \pm (1 - i)$

$w_1 = 2i + (1 - i) = 1 + i$

$w_2 = 2i - (1 - i) = -1 + 3i$

$z^3 = (yi)^3 = y^3 i^3 = y^3(-i) = -y^3 i$

$a = 0,\quad b = -y^3$

$2(w_1 + w_2) = 2\left((1+i) + (-1+3i)\right) = 2(4i) = 8i$

$8i = 8\,\text{cis}\,90°$

$z = \sqrt[3]{8}\,\text{cis}\!\left(\dfrac{90° + 360°k}{3}\right) = 2\,\text{cis}(30° + 120°k),\quad k = 0,1,2$

$z_1 = 2\,\text{cis}\,30° = 2\left(\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right) = \sqrt{3} + i$

$z_2 = 2\,\text{cis}\,150° = 2\left(-\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right) = -\sqrt{3} + i$

$z_3 = 2\,\text{cis}\,270° = 2(0 - i) = -2i$

$z_1 = \sqrt{3} + i \;\to\; (\sqrt{3},\, 1)$

$z_2 = -\sqrt{3} + i \;\to\; (-\sqrt{3},\, 1)$

$z_3 = -2i \;\to\; (0,\, -2)$

$\sqrt{0^2 + d^2} = 2$

$|d| = 2$