# If x = 5 - 5 ^ ( 2/3) - 5 ^ (1/3), prove that  x ^3 - 15x^2 + 60 x - 20 = 0?

May 8, 2018

Please see a Proof in the Explanation.

#### Explanation:

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

$\therefore 5 - x = \left({5}^{\frac{2}{3}} + {5}^{\frac{1}{3}}\right) \ldots \ldots \ldots \ldots \left(\ast\right)$.

$\therefore {\left(5 - x\right)}^{3} = {\left({5}^{\frac{2}{3}} + {5}^{\frac{1}{3}}\right)}^{3}$.

$\therefore {5}^{3} - {x}^{3} - 3 \left(5\right) \left(x\right) \left(5 - x\right) = {\left({5}^{\frac{2}{3}}\right)}^{3} + {\left({5}^{\frac{1}{3}}\right)}^{3}$

$+ 3 \times {5}^{\frac{2}{3}} \times {5}^{\frac{1}{3}} \left({5}^{\frac{2}{3}} + {5}^{\frac{1}{3}}\right) , i . e . ,$

$125 - {x}^{3} - 75 x + 15 {x}^{2} = {5}^{2} + {5}^{1} + 3 \times 5 \times \left(5 - x\right) \ldots \left[\because , \left(\ast\right)\right]$

$\therefore 125 - 75 x + 15 {x}^{2} - {x}^{3} = 30 + 75 - 15 x$.

$\therefore 20 - 60 x + 15 {x}^{2} - {x}^{3} = 0 ,$ as desired!