Suppose that positive real numbers $a$ and $b$ satisfy
$$a ^ { 2 } = 3 + \sqrt { 5 } , \quad b ^ { 2 } = 3 - \sqrt { 5 } .$$
Let $c$ be the fractional portion of $a + b$. We are to find the value of $\frac { 1 } { c } - c$.
(1) We see that $( a b ) ^ { 2 } = \mathbf { N }$ and $( a + b ) ^ { 2 } = \mathbf { O P }$.
(2) Since $\mathbf { Q }$ $< a + b < \mathbf { Q } + 1$, the value of $c$ is $\sqrt { \mathbf { R S } } - \mathbf { T }$.
Thus we obtain $\frac { 1 } { c } - c = \mathbf { U }$.