Problem 15. Solve the equation

$$
\frac{5}{\sqrt{1-x^{2}}}=\frac{1}{x}+\frac{85}{12}
$$

104
The positive function $P(x)=\frac{5}{\sqrt{1-x^{2}}}$ is defined on $(-1 ; 0)$ and $(0 ; 1)$. The positive function $F(x)=\frac{1}{x}+\frac{85}{12}$ is considered on $\left(-1 ;-\frac{12}{85}\right)$ and $(0 ; 1)$. To form a hypothesis about the number of roots of the equation, we construct tables of the functions $P(x)$ and $F(x)$:

| $x$ | $P(x)$ | $F(x)$ | $x$ | $P(x)$ | $F(x)$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| -1 | $\infty$ | 6.08 | 0 | 5 | $\infty$ |
| $-0.9$ | 11.4 | 5.97 | 0.1 | 5.02 | 17.1 |
| $-0.8$ | 8.33 | 5.83 | 0.2 | 5.10 | 12.1 |
| $-0.7$ | 7 | 5.65 | 0.3 | 5.24 | 10.4 |
| $-0.6$ | 6.25 | 5.41 | 0.4 | 5.46 | 9.58 |
| $-0.5$ | 5.77 | 5.08 | 0.5 | 5.77 | 9.08 |
| $-0.4$ | 5.46 | 4.58 | 0.6 | 6.25 | 8.75 |
| $-0.3$ | 5.24 | 3.75 | 0.7 | 7 | 8.51 |
| $-0.2$ | 5.10 | 2.08 | 0.8 | $8 \frac{1}{3}$ | $8 \frac{1}{3}$ |
| $-\frac{12}{85}$ | 5.05 | 0 | 1 | $\infty$ | 8.08 |

The functions $P(x)$ and $F(x)$ on $\left(-1 ;-\frac{12}{85}\right)$ are decreasing. The "staircase" method easily proves that the equation (1) has no roots on this interval. On $(0 ; 1)$, the continuous function $P(x)$ is increasing, and the function $F(x)$ is decreasing. The number 0.8 is the only root of the equation (1).