Show-that-tan-5pi-12-is-a-solution-of-this-equation-x-3-3x-2-3x-1-0-

Question Number 87382 by mathocean1 last updated on 04/Apr/20

Show that \tan \frac{5 π}{12} is a solution of this

equation : x^{3} - 3 x^{2} - 3 x + 1 = 0

Answered by ajfour last updated on 04/Apr/20

x^{3} + 1 = 3 x (x + 1)

i f x \neq - 1

x^{2} - x + 1 = 3 x

x^{2} - 4 x + 1 = 0

x = 2 \pm \sqrt{3}

\tan \frac{5 π}{12} = \tan 75 ° = \tan (30 ° + 45 °)

= \frac{\frac{1}{\sqrt{3}} + 1}{1 - \frac{1}{\sqrt{3}}} = \frac{1 + \sqrt{3}}{\sqrt{3} - 1} = \frac{4 + 2 \sqrt{3}}{2} = 2 + \sqrt{3} .

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