x-3-x-2-x-c-0-find-x-

Question Number 175095 by Gbenga last updated on 18/Aug/22

x^{3} + x^{2} + x + c = 0

find x

Answered by MJS_new last updated on 18/Aug/22

x = z - \frac{1}{3}

z^{3} + \frac{2}{3} z + (c - \frac{7}{27}) = 0

\Rightarrow

z_{1} = u + v

z_{2} = ω u + ω^{2} v

z_{3} = ω^{2} u + ω v

ω = - \frac{1}{2} + \frac{\sqrt{3}}{2} i

u = \sqrt[3]{- \frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}} \land v = - \sqrt[3]{\frac{q}{2} \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}}}

p = \frac{2}{3} \land q = c - \frac{7}{27}

D = \frac{p^{3}}{27} + \frac{q^{2}}{4} = \frac{c^{2}}{4} - \frac{7 c}{54} + \frac{1}{36} > 0 \forall c \in R

do the rest yourself

Commented by Gbenga last updated on 19/Aug/22

Commented by Tawa11 last updated on 19/Aug/22

Great sir

Facebook Tweet Pin