How to solve this equation given another equation with the same variable

Question

Please help me. I've tried for quite a while but I do not know how to get the correct expression. The answer is 2 according to the markscheme.

Answer 1

Note that $$a^3-\frac {1}{a^3} = (a-\frac {1}{a})^3+3(a-\frac {1}{a})=8+6=14$$

Also $$a^2+\frac{1}{a^2} = (a-\frac {1}{a})^2+2=6$$

Thus $$a^3-\frac {1}{a^3} -2(a^2+\frac{1}{a^2})=14-2(6)=2$$

Answer 2

$ a - \frac{1}{a} = 2$ implies that $a^{2} - 1 = 2a$; which equates to the quadratic equation $a^{2} - 2a - 1 = 0$, having solutions $a = 1 \pm \sqrt{2}$. Now, substitute into the other equation and evaluate.