Practice Mathathon Problem #001

$\begin{aligned} x & = 2 + 2^\frac 23 + 2^\frac 13 \\ & = 2^\frac 13 (2^\frac 23 + 2^\frac 13 + 1) \\ \implies x & = 2^\frac 13 (x-1) & \small \blue{\text{Cubing both sides}} \\ x^3 & = 2(x-1)^3 \\ & = 2(x^3 - 3x^2 + 3x - 1) \\ & = 2x^3 - 6x^2 + 6x - 2 \\ \implies x^3 - 6x^2 + 6x & = \boxed 2 \end{aligned}$

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

Cubing both sides,

LHS = (x-2)^3 = x^3 - (2)^3 -3(2)(x) (x-2) = x^3 - 6x(x-2) - 8 = x^3 - 6x^2 + 12x - 8

RHS = (2^(2/3) + 2^(1/3))^3 = (2^(2/3))^3 + (2^(1/3)^3 + 3 (2)^(2/3)2^(1/3) (2^(2/3) + 2^(1/3)) = 4 + 2 + 3 * 2(2^(2/3) + 2^(1/3)) = 6 + 6(2^(2/3) + 2^(1/3)) ....(1)

The expression that requires evaluation is : x^3 - 6x^2 + 6x = (x^3 - 6x^2 + 12x - 8) - 6x + 8 = LHS - 6x + 8 = RHS - 6x + 8 ...(2)

Plugging in values of RHS from (1) and the given expression for x, we get,

x^3 - 6x^2 + 6x = 6 + 6(2^(2/3) + 2^(1/3)) - 6(2 + 2^(2/3) + 2^(1/3)) + 8 = 6 - 12 + 6(2^(2/3) + 2^(1/3)) - 6(2^(2/3) + 2^(1/3)) + 8 = -6 + 8 = 2

If we input $x^{3}-6x^{2}+6x$ , and $x=2+2^{\frac{2}{3}}+2^{\frac{1}{3}}$ into Desmos, we get

Most Common Solution -

$\begin{aligned} x &= 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}} \\ x - 2 &= 2^{\frac{2}{3}} + 2^{\frac{1}{3}} \\ (x - 2)^{3} &= (2^{\frac{2}{3}} + 2^{\frac{1}{3}})^{3} \\ x^{3} -6x^{2} + 12x - 8 &= 4 + 2 + 6(2^{\frac{2}{3}} + 2^{\frac{1}{3}}) \\ x^{3} - 6x^{2} + 6x &= 6 + 6(x-2) - 6x + 8 \\ x^{3} - 6x^{2} + 6x &= \boxed{2} \end{aligned}$