205. Disgraced Umbugio. Recently, Professor Euclid Paracelsus Bombast Umbugio was walking around puffed up with pride, as he managed to guess the root of a fourth-degree equation that resulted after eliminating radicals from the equation

$$
x=\left(x-\frac{1}{x}\right)^{\frac{1}{2}}+\left(1-\frac{1}{x}\right)^{\frac{1}{2}}
$$

Disgrace the professor and solve this equation using equations no higher than the second degree.
205. Let $b=\left(x-\frac{1}{x}\right)^{\frac{1}{2}}$ and $a=\left(1-\frac{1}{x}\right)^{\frac{1}{2}}$; then

$$
x=a+b
$$

and since $n \neq 0$

$$
b-a=\frac{b^{2}-a^{2}}{b+a}=\frac{x-1}{x}=1-\frac{1}{x}
$$

Adding (1) and (2), we get

$$
2 b=x-\frac{1}{x}+1=b^{2}+1
$$

from which $b=1$.

Therefore,

$$
x-\frac{1}{x}=1, \quad x^{2}-x-1=0 \quad \text { and } \quad x=\frac{1 \pm \sqrt{5}}{2}
$$

However, the original equation is satisfied only by

$$
x=\frac{1+\sqrt{5}}{2}
$$

[P. M. Anselone and S. Cook, A. M. M., 62, 728 (December 1955).]

[By the way, how did Professor Umbudjo solve this equation? Ped. $\left.{ }^{*}\right]$