Solve-x-3-18x-32-0-

Question Number 9378 by tawakalitu last updated on 03/Dec/16

Solve :

x^{3} - 18 x - 32 = 0

Answered by mrW last updated on 03/Dec/16

a = 1, b = 0, c = - 18, d = - 32

Δ = b^{2} - 3 ac = - 3 \times 1 \times (- 18) = 54

k = \frac{9 abc - {2 b}^{3} - {27 a}^{2} d}{2 \sqrt{∣ Δ ∣^{3}}} = \frac{- 27 \times (- 32)}{2 \sqrt{54^{3}}} = \frac{4 \sqrt{6}}{9} \approx 1.09

since Δ > 0 and ∣ k ∣> 1, there is a single root :

x_{1} = \frac{\sqrt{54}}{3} \times (^{3} \sqrt{\frac{4 \sqrt{6}}{9} + \sqrt{{(\frac{4 \sqrt{6}}{9})}^{2} - 1}} +^{3} \sqrt{\frac{4 \sqrt{6}}{9} - \sqrt{{(\frac{4 \sqrt{6}}{9})}^{2} - 1}})

= \sqrt{6} \times (^{3} \sqrt{\frac{4 \sqrt{6} + \sqrt{15}}{9}} +^{3} \sqrt{\frac{4 \sqrt{6} - \sqrt{15}}{9}})

\approx 4.94662127666289

Commented by tawakalitu last updated on 03/Dec/16

i really appreciate sir .