Question

If$\frac{\left(\sqrt{5}-1\right)}{\left(\sqrt{5}+1\right)}+\frac{\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)}=\left(a+b\sqrt{5}\right)$, find the values of a and b.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \frac{\sqrt{5}-1}{\sqrt{5}+1}+\frac{\sqrt{5}+1}{\sqrt{5}-1} \\ =\frac{{\left(\sqrt{5}-1\right)}^2+{\left(\sqrt{5}+1\right)}^2}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)} \\ =\frac{5+1-2\sqrt{5}+5+1+2\sqrt{5}}{{\left(\sqrt{5}\right)}^2-1} \\ =\frac{12}{4} \\ =3 \\ \mathrm{We}\mathrm{know}: \\ a+b\sqrt{5}=3 \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ a=3\mathrm{and}b=0 \end{gathered}$$