solve : ( x^( 2) +x −6)^( 3) + (7x^( 2) −9x −2)^( 3) −512(x^2 −x−1)^( 3) =0 x = ?


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Question Number 158156 by mnjuly1970 last updated on 31/Oct/21

$s o l v e :$ ${(x^{2} + x - 6)}^{3} + {(7 x^{2} - 9 x - 2)}^{3} - 512 {(x^{2} - x - 1)}^{3} = 0$ $x = ?$
	Answered by som(math1967) last updated on 31/Oct/21

	${(x^{2} + x - 6)}^{3} + {(7 x^{2} - 9 x - 2)}^{3} + {(- 8 x^{2} + 8 x + 8)}^{3} = 0$ $l e t x^{2} + x - 6 = a, 7 x^{2} - 9 x - 2 = b$ $- 8 x^{2} + 8 x + 8 = c$ $∴ a + b + c = 0$ $∴ a^{3} + b^{3} + c^{3} = 3 a b c$ $∴ 3 (x^{2} + x - 6) (7 x^{2} - 9 x - 2) (- 8 x^{2} + 8 x + 8) = 0$ $e i t h e r (x^{2} + x - 6) = 0$ $x = - 3, 2$ $7 x^{2} - 9 x - 2 = 0$ $x = \frac{9 \pm \sqrt{81 + 56}}{14} = \frac{9 \pm \sqrt{137}}{14}$ $- 8 x^{2} + 8 x + 8 = 0$ $- x^{2} + x + 1 = 0$ $x = \frac{- 1 \pm \sqrt{1 + 4}}{- 2} = \frac{- 1 \pm \sqrt{5}}{- 2}$
	Commented by mnjuly1970 last updated on 31/Oct/21

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