146 1) $$5^{2x} - 5^x – 600 = 0$$; 2) $$9^x - 3^x - 6 = 0$$.

$$5^{2x} - 5^x – 600 = 0$$

$$(5^x)^2 - 5^x - 600 = 0$$

Пусть $$t = 5^x$$, тогда

$$t^2 - t - 600 = 0$$

$$D = (-1)^2 - 4 \cdot 1 \cdot (-600) = 1 + 2400 = 2401 = 49^2$$

$$t_1 = \frac{1 + 49}{2} = \frac{50}{2} = 25$$

$$t_2 = \frac{1 - 49}{2} = \frac{-48}{2} = -24$$

$$5^x = 25$$ или $$5^x = -24$$

$$5^x = 5^2$$

$$x = 2$$

Ответ: $$x = 2$$

$$9^x - 3^x - 6 = 0$$

$$(3^2)^x - 3^x - 6 = 0$$

$$(3^x)^2 - 3^x - 6 = 0$$

Пусть $$t = 3^x$$, тогда

$$t^2 - t - 6 = 0$$

$$D = (-1)^2 - 4 \cdot 1 \cdot (-6) = 1 + 24 = 25 = 5^2$$

$$t_1 = \frac{1 + 5}{2} = \frac{6}{2} = 3$$

$$t_2 = \frac{1 - 5}{2} = \frac{-4}{2} = -2$$

$$3^x = 3$$ или $$3^x = -2$$

$$3^x = 3^1$$

$$x = 1$$

Ответ: $$x = 1$$