Question

a^2-4a-1=0 and a is not equal to 0,

find

1. a-1/a
2. a=1/a
3. a^2-1/a^2

Answer 1

$$\begin{gathered} \left(i\right) \\ {a}^2-4a-1=0 \\ \Rightarrow a\left(a-4\right)=1 \\ \Rightarrow a-4=\frac{1}{a} \\ \Rightarrow a-\frac{1}{a}=4 \\ \left(ii\right) \\ {\left(a+\frac{1}{a}\right)}^2={\left(a-\frac{1}{a}\right)}^2+4 \\ ={\left(4\right)}^2+4 \\ =16+4 \\ =20 \\ \Rightarrow a+\frac{1}{a}=\sqrt{20} \\ \Rightarrow a+\frac{1}{a}=2\sqrt{5} \\ \left(iii\right) \\ {a}^2-\frac{1}{a^2}=\left(a+\frac{1}{a}\right)\left(a-\frac{1}{a}\right)=4\times 2\sqrt{5}=8\sqrt{5} \\ \Rightarrow a^2-\frac{1}{a^2}=8\sqrt{5} \end{gathered}$$