Решение

$$d_1 + f_1 = 90 \text{ см}$$
$$x = 30 \text{ см}$$
$$\frac{1}{F} = \frac{1}{d_1} + \frac{1}{f_1}$$
$$\frac{1}{F} = \frac{1}{d_1 + x} + \frac{1}{f_1 - x}$$
$$\frac{1}{d_1} + \frac{1}{f_1} = \frac{1}{d_1 + x} + \frac{1}{f_1 - x}$$
$$\frac{f_1 + d_1}{d_1 f_1} = \frac{(f_1 - x) + (d_1 + x)}{(d_1 + x)(f_1 - x)}$$
$$\frac{1}{d_1 f_1} = \frac{1}{(d_1 + x)(f_1 - x)}$$
$$d_1 f_1 = (d_1 + x)(f_1 - x)$$
$$d_1 f_1 = d_1 f_1 - d_1 x + x f_1 - x^2$$
$$0 = -d_1 x + x f_1 - x^2$$
$$x^2 = x(f_1 - d_1)$$
$$x = f_1 - d_1$$
$$\frac{1}{F} = \frac{1}{d_1} + \frac{1}{f_1}$$
$$x = f_1 - d_1$$
$$F = \frac{d_1 f_1}{d_1 + f_1}$$
$$F = \frac{(d_1 + f_1)^2 - x^2}{4(d_1 + f_1)}$$
$$F = \frac{90^2 - 30^2}{4 \cdot 90} = 20 \text{ см}$$