Math Olympiad | A Nice Rational Equation | VIJAY Maths

[@adribber]

x + x² + x³ + 1/x + 1/x² + 1/x³ = 6
(x + 1/x) + (x² + 1/x²) + (x³ + 1/x³) = 6
(x + 1/x) + (x + 1/x)² - 2 + (x + 1/x)³ - 3(x + 1/x) = 6
(x + 1/x)³ + (x + 1/x)² - 2(x + 1/x) - 8 = 0
Let, k = x + 1/x
k³ - 2³ + k² - 2k = 0
(k - 2)(k² + 2k + 4) + k(k - 2) = 0
(k - 2)(k² + 2k + 4 + k) = 0
(k - 2)(k² + 3k + 4) = 0
k - 2 = 0 , k² + 3k + 4 = 0
From, k² + 3k + 4 = 0
D = 3² - 4*1*4 = 9 - 16 = - 7 < 0 Rejected
From, k - 2 = 0
Recall, k = x + 1/x
x + 1/x = 2
x² - 2x + 1 = 0
(x - 1)² = 0
∴ x = 1

[@sandytanner9333]

1

[@yakupbuyankara5903]

X=1

[@RealQinnMalloryu4]

{x+x ➖}+{x^2+x^2 ➖}+{x^3+x^3 ➖}+{1+1 ➖ }/{x+x ➖}+{1+1 ➖ }/{x^2+x^2 ➖}+{1+1 ➖ }/{x^3+x^3 ➖}={x^2+x^4+x^6}+{2/x^2+2/x^4+2/x^6}={x^12+6}/x^12=6x^12/x^12=6x^1 2^3x^1 2^1x^1 2x^1 (x ➖ 2x+1).