Answer

Solution

1. 1

   From $x^2 - 3x + 1 = 0$ (and $x \ne 0$, since the constant term is non-zero), divide by $x$:

2. 2
   $$x + \frac{1}{x} = 3.$$
3. 3

   Square:

4. 4
   $$x^2 + \frac{1}{x^2} = \left(x + \tfrac{1}{x}\right)^2 - 2 = 9 - 2 = 7.$$
5. 5

   Square again:

6. 6
   $$x^4 + \frac{1}{x^4} = \left(x^2 + \tfrac{1}{x^2}\right)^2 - 2 = 49 - 2 = 47.$$

Answer structure / marking notes

Content note