solve-x-2-x-6-3-7x-2-9x-2-3-512-x-2-x-1-3-0-x-

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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