x^2 - 13x + 1 = 0
The sum of the roots =13
The product of the roots = 1
But....if the product of the roots is 1.... we can call one root x and the other 1/x
And the sum of the roots = x + 1/x = 13
So
(x)^2 - 13(x) + 1 = 0
(1/x)^2 - 13 (1/x) + 1 = 0 add these
(x)^2 + (1/x)^2 - 13 ( x + 1/x) + 2 = 0
(x)^2 + (1/x)^2 -13 (13) + 2 = 0
(x)^2 + (1/x)^2 - 169 + 2 = 0
(x)^2 + (1/x)^2 -167 = 0
(x)^2 + (1/x)^2 = 167 square both sides
(x)^4 + 2 (x)^2 * (1/x)^2 + (1/x)^2 = 27889 simplify
x^4 + 2 (x^2/x^2) + 1/x^4 = 27889
(x)^4 + 2 + (1/x)^4 + 2 = 27889 subtract 2 from both sides
x^2 + 1/x^4 = 27887
![cool cool](/img/emoticons/smiley-cool.gif)