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Question Number 14658 by tawa tawa last updated on 03/Jun/17

Given that:  x = ((√2) + 1)^(1/3)  − ((√2) − 1)^(1/3)   Show that ,     x^3  + 3x = 2

$$\mathrm{Given}\:\mathrm{that}: \\ $$$$\mathrm{x}\:=\:\left(\sqrt{\mathrm{2}}\:+\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \:−\:\left(\sqrt{\mathrm{2}}\:−\:\mathrm{1}\right)^{\mathrm{1}/\mathrm{3}} \\ $$$$\mathrm{Show}\:\mathrm{that}\:,\:\:\:\:\:\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{3x}\:=\:\mathrm{2} \\ $$

Answered by Tinkutara last updated on 03/Jun/17

x^3  = (√2) + 1 − (√2) + 1 − 3(1)(x)  [Using (a − b)^3  = a^3  − b^3  − 3ab(a − b)]  x^3  + 3x = 2

$${x}^{\mathrm{3}} \:=\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}\:−\:\sqrt{\mathrm{2}}\:+\:\mathrm{1}\:−\:\mathrm{3}\left(\mathrm{1}\right)\left({x}\right) \\ $$$$\left[\mathrm{Using}\:\left({a}\:−\:{b}\right)^{\mathrm{3}} \:=\:{a}^{\mathrm{3}} \:−\:{b}^{\mathrm{3}} \:−\:\mathrm{3}{ab}\left({a}\:−\:{b}\right)\right] \\ $$$${x}^{\mathrm{3}} \:+\:\mathrm{3}{x}\:=\:\mathrm{2} \\ $$

Commented by tawa tawa last updated on 03/Jun/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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