x^3 −(3/4)x+(1/8)=0 x=?


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Question Number 176995 by blackmamba last updated on 29/Sep/22

$x^{3} - \frac{3}{4} x + \frac{1}{8} = 0$ $x = ?$
	Answered by mr W last updated on 29/Sep/22

	$Δ = {(\frac{1}{16})}^{2} + {(- \frac{1}{4})}^{3} = - \frac{3}{256} < 0$ $\Rightarrow t h r e e r e a l r o o t s!$ $x = 2 \sqrt{\frac{1}{4}} \sin (\frac{1}{3} \sin^{- 1} \frac{\frac{1}{16}}{\frac{1}{4} \sqrt{\frac{1}{4}}} + \frac{2 k π}{3})$ $x = \sin (\frac{1}{3} \sin^{- 1} \frac{1}{2} + \frac{2 k π}{3})$ $x = \sin (\frac{π}{18} + \frac{2 k π}{3}), k = 0, 1, 2$ $x = \sin 10^{°}, \sin 130 °, \sin 250 °$
	Commented by mr W last updated on 29/Sep/22

	$s e e Q 155945$
	Commented by Tawa11 last updated on 02/Oct/22

	$Great sir$
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