(1/x) + (1/x)² = 6 → let: y = 1/x

y + y² = 6

y² + y + (1/4) = 6 + (1/4)

[y + (1/2)]² = 25/4

y + (1/2) = ± 5/2

y = - (1/2) ± (5/2)

y = (- 1 ± 5)/2

y = (- 1 - 5)/2 = - 3 → x = - 1/3

y = (- 1 + 5)/2 = 2 → x = 1/2

→ Solution = { - 1/3 ; 1/2 }

1/x + (1/x)² = 6

1/x + 1/x^2 = 6

( 1X + 1 )/X^2 = 6

1 X + 1 = 6 X^2

6 X^2 - X - 1 = 0

X = ( 1 +- Raiz 1^2 + 4 * 6 * 1 ) / 12

X = ( 1 +- 5 ) / 12

X = ( 1 + 5 ) / 12

X = 6/12

X = 1/2

X´ = ( 1 - 5 ) / 12

X´= - 4/12

X´= - 1/3

.....................................................

1/1/2 + 1/1/4 = 2 + 4

1/x + (1/x)² = 6

1/x + 1/x² = 6

Multiplico todo por x²:

x + 1 = 6x²

6x² - x - 1 = 0

Las dos respuestas posibles son:

x1 = 1/2

x2 = -1/3