Finding value of an expression If $x^2-3x-1=0$ then find the value of $(x^6+1)/x^3$ I tried to solve the quadratic but it became too complicated any way of doing this without a calculator

$$x-\frac1x=3.$$
Then
$$\left(x-\frac1x\right)^2+3=x^2+1+\frac1{x^2}=12.$$
Then
$$\left(x^2+1+\frac1{x^2}\right)\left(x-\frac1x\right)=x^3-\frac1{x^3}=36.$$
Finally,
$$\left(x^3+\frac1{x^3}\right)^2=\left(x^3-\frac1{x^3}\right)^2+4=1300.$$
