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Question Number 21296 by NECx last updated on 19/Sep/17

Commented by NECx last updated on 19/Sep/17

$please help me solve this . It was$ $solved before but i cant assess the$ $answer again .$ $please help$
Commented by NECx last updated on 20/Sep/17

$i think then it was solved by$ $smalso 334 . . . . something like that$
Commented by NECx last updated on 20/Sep/17

$please help$
	Answered by Joel577 last updated on 20/Sep/17

	${2 \sin}^{- 1} (x \sqrt{6}) + \sin^{- 1} (4 x) = \frac{π}{2}$ $\sin^{- 1} x \sqrt{6} = α \to \sin α = x \sqrt{6}$ $\sin^{- 1} 4 x = β \to \sin β = 4 x$ $\sin ({2 \sin}^{- 1} x \sqrt{6} + \sin^{- 1} 4 x) = \sin \frac{π}{2}$ $\sin (2 α + β) = 1$ $\sin 2 α \cos β + \cos 2 α \sin β = 1$ $2 \sin α \cos α \cos β + (1 - {2 \sin}^{2} α) \sin β = 1$ $2 \sin α \cos α \cos β = 2 . x \sqrt{6} . \sqrt{1 - 6 x^{2}} . \sqrt{1 - 16 x^{2}}$ $= 2 x \sqrt{6 (1 - 6 x^{2}) (1 - 16 x^{2})}$ $(1 - {2 \sin}^{2} α) \sin β = (1 - 12 x^{2}) 4 x$ $= 4 x - 48 x^{3}$ $2 x \sqrt{6 (1 - 6 x^{2}) (1 - 16 x^{2})} + 4 x - 48 x^{3} = 1$ $4 x^{2} \sqrt{576 x^{4} - 132 x^{2} + 6} = 48 x^{3} - 4 x + 1$ $4 x^{2} (576 x^{4} - 132 x^{2} + 6) = 2304 x^{6} - 384 x^{4} + 96 x^{3} + 16 x^{2} - 8 x + 1$ $2304 x^{6} - 528 x^{4} + 24 x^{2} = 2304 x^{6} - 384 x^{4} + 96 x^{3} + 16 x^{2} - 8 x + 1$ $144 x^{4} + 96 x^{3} - 8 x^{2} - 8 x + 1 = 0$ $If we plot it into Cartecian plane, it shows$ $x = - \frac{1}{2} or x = \frac{1}{6}$
	Commented by Joel577 last updated on 20/Sep/17

	$If someone have shorter ways to solve this,$ $please explain it to us . Thanks$
	Commented by NECx last updated on 20/Sep/17

	$thanks boss$
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