solve-x-3-3x-1-0-

Question Number 82582 by M±th+et£s last updated on 22/Feb/20

s o l v e

x^{3} - 3 x + 1 = 0

Commented by mr W last updated on 23/Feb/20

x^{3} + 3 (- 1) x + 2 (\frac{1}{2}) = 0

p = - 1

q = \frac{1}{2}

Δ = p^{3} + q^{2} = - 1 + \frac{1}{4} = - \frac{3}{4} < 0

\Rightarrow t h r e e r e a l r o o t s

x = 2 \sqrt{- p} \sin (\frac{1}{3} \sin^{- 1} \frac{q}{\sqrt{- p^{3}}} + \frac{2 k π}{3})

= 2 \sin (\frac{1}{3} \sin^{- 1} \frac{1}{2} + \frac{2 k π}{3})

= 2 \sin (\frac{π}{18} + \frac{2 k π}{3}) (k = 0, 1, 2)

x_{1} = 2 \sin (\frac{π}{18}) = 0.347296

x_{2} = 2 \sin (\frac{π}{18} + \frac{2 π}{3}) = 1.532089

x_{3} = 2 \sin (\frac{π}{18} + \frac{4 π}{3}) = - 1.879385

Commented by M±th+et£s last updated on 23/Feb/20

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