Question

If $a=3-2\sqrt{2}$, find the value of $a^2-\frac{1}{a^2}$.

Answer 1

$$\begin{gathered} a=3-2\sqrt{2} \\ \Rightarrow a^2={\left(3-2\sqrt{2}\right)}^2 \\ \Rightarrow a^2=9+8-12\sqrt{2} \\ \Rightarrow a^2=17-12\sqrt{2}.....\left(1\right) \end{gathered}$$
$$\begin{gathered} \therefore \frac{1}{a^2}=\frac{1}{17-12\sqrt{2}} \\ \Rightarrow \frac{1}{a^2}=\frac{1}{17-12\sqrt{2}}\times \frac{17+12\sqrt{2}}{17+12\sqrt{2}} \\ \Rightarrow \frac{1}{a^2}=\frac{17+12\sqrt{2}}{{17}^2-{\left(12\sqrt{2}\right)}^2} \end{gathered}$$
$$\begin{gathered} \Rightarrow \frac{1}{a^2}=\frac{17+12\sqrt{2}}{289-288} \\ \Rightarrow \frac{1}{a^2}=17+12\sqrt{2}.....\left(2\right) \end{gathered}$$
Subtracting (2) from (1), we get
$$\begin{gathered} a^2-\frac{1}{a^2}=\left(17-12\sqrt{2}\right)-\left(17+12\sqrt{2}\right) \\ \Rightarrow a^2-\frac{1}{a^2}=17-12\sqrt{2}-17-12\sqrt{2} \\ \Rightarrow a^2-\frac{1}{a^2}=-24\sqrt{2} \end{gathered}$$
Thus, the value of $a^2-\frac{1}{a^2}$ is $-24\sqrt{2}$.