ax-3-bx-2-c-0-x-1-x-2-x-3-

Question Number 183202 by liuxinnan last updated on 23/Dec/22

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

x_{1} =

x_{2} =

x_{3} =

Answered by Frix last updated on 23/Dec/22

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

m = \frac{b}{a} \land n = \frac{c}{a}

x^{3} + {m x}^{2} + n = 0

x = t - \frac{m}{3}

t^{3} + \frac{m^{2}}{3} t + \frac{2 m^{3}}{27} + n = 0

p = \frac{m^{2}}{3} \land q = \frac{2 m^{3}}{27} + n

t^{3} + p t + q = 0

Depending on D = \frac{p^{3}}{27} + \frac{q^{2}}{4} \underset{<}{\overset{>}{=}} 0 you can use

{Cartano}^{'} s Formula or the Trigonometric

Formula :

D > 0 \Rightarrow

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

[you must take the real roots : \sqrt[3]{- r} = - \sqrt[3]{r}]

t_{1} = u + v

t_{2} = (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) u + (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) v

t_{3} = (- \frac{1}{2} - \frac{\sqrt{3}}{2} i) u + (- \frac{1}{2} + \frac{\sqrt{3}}{2} i) v

D = 0 \land p \neq 0 \land q \neq 0 \Leftrightarrow \frac{p^{3}}{27} = - \frac{q^{2}}{4} \Rightarrow

u = v = \sqrt[3]{- \frac{q}{2}}

t_{1} = 2 \sqrt[3]{- \frac{q}{2}} = \frac{3 q}{p}

t_{2} = t_{3} = - \sqrt[3]{- \frac{q}{2}} = - \frac{3 q}{2 p}

D < 0 \Rightarrow

t_{k} = \sqrt[3]{- \frac{4 p}{3}} \times \cos (\frac{2 π k + \cos^{- 1} (- \frac{q}{2} \times \sqrt{- \frac{27}{p^{3}}})}{3}) with k = 1, 2, 3

You must insert backwards

p = \frac{m^{2}}{3} \land q = \frac{2 m^{3}}{27} + n

x_{k} = t_{k} - \frac{m}{3}

m = \frac{b}{a} \land n = \frac{c}{a}

to get the demanded formula

Answered by mr W last updated on 23/Dec/22

w e c a n a l s o s o l v e f o r \frac{1}{x} i n s t e a d o f

f o r x .

\frac{1}{x^{3}} + \frac{b}{c x} + \frac{a}{c} = 0

Δ = \frac{1}{4} {(\frac{a}{c})}^{2} + \frac{1}{27} {(\frac{b}{c})}^{3}

\underset{―}{i f Δ > 0 :}

\frac{1}{x_{1}} = \sqrt[3]{Δ - \frac{a}{2 c}} - \sqrt[3]{Δ + \frac{a}{2 c}}

\frac{1}{x_{2, 3}} = - \frac{1}{2} (\sqrt[3]{Δ - \frac{a}{2 c}} - \sqrt[3]{Δ + \frac{a}{2 c}}) \pm (\sqrt[3]{Δ - \frac{a}{2 c}} + \sqrt[3]{Δ + \frac{a}{2 c}}) i

\underset{―}{i f Δ = 0 :}

\frac{1}{x_{1}} = - 2 \sqrt[3]{\frac{a}{2 c}}

\frac{1}{x_{2}} = \frac{1}{x_{3}} = \sqrt[3]{\frac{a}{2 c}}

\underset{―}{i f Δ < 0 :}

\frac{1}{x_{1, 2, 3}} = 2 \sqrt{- \frac{b}{3 c}} \sin [\frac{1}{3} \sin^{- 1} (\frac{3 a}{- 2 b} \sqrt{- \frac{3 c}{b}}) + \frac{2 k π}{3}] (k = 0, 1, 2)

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