If $x = 5 - 5^{\frac{2}{3}} - 5^{\frac{1}{3}}$ $x = 5 - 5 ^ ( 2/3) - 5 ^ (1/3)$ , prove that $x^{3} - 15 x^{2} + 60 x - 20 = 0$ $x ^3 - 15x^2 + 60 x - 20 = 0$ ?

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Answer 1 · 2018-05-08T11:16:33

Please see a Proof in the Explanation.

$x = 5 - 5^{\frac{2}{3}} - 5^{\frac{1}{3}}$ $x=5-5^(2/3)-5^(1/3)$ .

$:. 5-x=(5^(2/3)+5^(1/3))............(ast)$ .

$:. (5-x)^3=(5^(2/3)+5^(1/3))^3$ .

$:. 5^3-x^3-3(5)(x)(5-x)=(5^(2/3))^3+(5^(1/3))^3$

$+3xx5^(2/3)xx5^(1/3)(5^(2/3)+5^(1/3)), i.e.,$

$125-x^3-75x+15x^2=5^2+5^1+3xx5xx(5-x)...[because, (ast)]$

$:. 125-75x+15x^2-x^3=30+75-15x$ .

$:. 20-60x+15x^2-x^3=0,$ as desired!

If x = 5 - 5 ^ ( 2/3) - 5 ^ (1/3)x=5−523−513x = 5 - 5 ^ ( 2/3) - 5 ^ (1/3), prove that x ^3 - 15x^2 + 60 x - 20 = 0x3−15x2+60x−20=0 x ^3 - 15x^2 + 60 x - 20 = 0?