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Question Number 185198 by ajfour last updated on 18/Jan/23

Commented by ajfour last updated on 18/Jan/23

$I f z = r + x i + y j$ $z^{2} = r^{2} - x^{2} - y^{2} + 2 r (x i + y j)$ $+ x y (i j + j i)$ $l e t j i = i a n d i j = j$ $\Rightarrow z^{2} = r^{2} - x^{2} - y^{2} + x (2 r + y) i$ $+ y (2 r + x) j$ $z^{3} = r (r^{2} - x^{2} - y^{2}) + r x (2 x + y) i$ $+ r y (2 r + x) j + x (r^{2} - x^{2} - y^{2}) i$ $- x^{2} (2 r + y) + x y (2 r + x) j$ $+ y (r^{2} - x^{2} - y^{2}) j + x y (2 r + y) i$ $- y^{2} (2 r + x)$ $z^{3} = r^{3} - r (3 x^{2} + 3 y^{2}) - x y (x + y)$ $+ i (r^{2} - x^{2} + 2 r y) x + j (r^{2} - y^{2} + 2 r x) y$ $★$
Commented by ajfour last updated on 18/Jan/23

$N o w i f p^{3} - p - c = 0 w i t h c < \frac{2}{3 \sqrt{3}}$ $t h e n p = {(\frac{c}{2} + i \sqrt{\frac{1}{27} - \frac{c^{2}}{4}})}^{1 / 3}$ $+ {(\frac{c}{2} - j \sqrt{\frac{1}{27} - \frac{c^{2}}{4}})}^{1 / 3}$ $f o r c = \frac{1}{3}$ $p = {(\frac{1}{6} + \frac{i}{6 \sqrt{3}})}^{1 / 3} + {(\frac{1}{6} - \frac{j}{6 \sqrt{3}})}^{1 / 3}$ $p \sqrt{3} = {(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{1 / 3} + {(\frac{\sqrt{3}}{2} - \frac{j}{2})}^{1 / 3}$ $= {(w^{3})}^{1 / 3} + {(z^{3})}^{1 / 3}$ $l e t w = r_{1} + x i + y j$ $z = r_{2} - x i - y j$ $p \sqrt{3} = w + z = r_{1} + r_{2}$
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