A Nice Rational Equation | Algebra Challenge | Math Olympiad Preparation 

Let x-1 = t and t^2 = a, which is positive. Then, the equation reduces to 7a^3 +100a^2 +80a=0 > a=0 or 7a^2+100a+80=0. The latter has no positive roots. So, a=0 > x=1.

t>0と設定しましたよね。t=0としてはいけないのでは？

X=1.

Math Olympiad Preparation: [(x + 1)⁶ + (x - 3)⁶]/[x⁶ + (x - 2)⁶] = 2⁶, x ϵ R; x =? Let: y = x - 1; x + 1 = y + 2, x - 3 = y - 2, x = y + 1, x - 2 = y - 1, y ϵ R [(x + 1)⁶ + (x - 3)⁶]/[x⁶ + (x - 2)⁶] = [(y + 2)⁶ + (y - 2)⁶]/[(y + 1)⁶ + (y - 1)⁶] = 2⁶ (y ± a)⁶ = y⁶ ± 6ay⁵ + 15a²y⁴ ± 20a³y³ + 15a⁴y² ± 6a⁵y + a⁶ (y + a)⁶ + (y - a)⁶ = 2(y⁶ + 15a²y⁴ + 15a⁴y² + a⁶) a = 2; (y + 2)⁶ + (y - 2)⁶ = 2[y⁶ + 15(2²)y⁴ + 15(2⁴)y² + 2⁶] = 2(y⁶ + 60y⁴ + 240y² + 64) a = 1; (y + a)⁶ + (y - a)⁶ = 2(y⁶ + 15y⁴ + 15y² + 1) [2(y⁶ + 60y⁴ + 240y² + 64)]/[2(y⁶ + 15y⁴ + 15y² + 1)] = 2⁶ y⁶ + 60y⁴ + 240y² + 64 = 2⁶(y⁶ + 15y⁴ + 15y² + 1) = 64y⁶ + 960y⁴ + 960y² + 64 63y⁶ + 900y⁴ + 720y² = 0, 9y²(7y⁴ + 100y² + 80) = 0; y ϵ R 7y⁴ + 100y² + 80 = 0, y² = (- 100 ± √7760)/14 = (- 100 + 88.09)/14 < 0, Rejected The calculation was achieved on a smartphone with a standard calculator app y = 0 = x - 1, x = 1; Double root Answer check: [(x + 1)⁶ + (x - 3)⁶]/[x⁶ + (x - 2)⁶] = [2⁶ + (- 2)⁶]/[1 + (- 1)⁶] = 2(2⁶)/2 = 2⁶; Confirmed Final answer: x = 1

X=1

6x^6+(x^6 ➖ 729l/{x^6+x^6 ➖}(x^6 ➖ 64)={ 6x^6+723}/{x^12+58}= 729x^6/58x^12=150.1.1x^2 10^10^5^10 1^1.1^1x^2 2^52^5^52^5x^2 1^1^1^1^1^1^1x^2 1x^2 (x ➖ 2x+1).

Slightly disappointed with the approach of interpretation (explanation). Lack of straightforwardness and lack of clarity. Lazy approach! I seek forgiveness for harsh comment!

x= 1