Question

Find the roots of the the equation x+1/x3+1 - x-2/x2-4 = 0

Answer 1

Answer:

The roots are   1 + √2   and   1 - √2

Step-by-step explanation:

First, notice that

  x³ + 1 = ( x + 1 ) ( x² - x + 1 )

so

 ( x + 1 ) / ( x³ + 1 ) = 1 / ( x² - x + 1 ).

Also

  x² - 4 = ( x - 2 ) ( x + 2 )

so

  ( x - 2 ) / ( x² - 4 ) = 1 / ( x + 2 ).

Thus the equation can be rewritten as

1 / ( x² - x + 1 )  -  1 / ( x + 2 ) = 0

Multiplying both sides by ( x + 2 ) ( x² - x + 1 ), this becomes

( x + 2 ) - ( x² - x + 1 ) = 0

=> x + 2 - x² + x - 1 = 0

=> x² - 2x = 1

=> x² - 2x + 1 = 1 + 1 = 2

=> ( x - 1 )² = 2

=> x = 1 ± √2