Example 6. Solve the equation

$$
\sqrt{5-x}\left(3^{x^{2}-7.2 x+3.9}-9 \sqrt{3}\right)=0
$$
Solution. The domain of permissible values of equation (1) is determined by the condition $5-x \geqslant 0$, i.e., $x \leqslant 5$. For such values of $x$, equation (1) is equivalent to the system of equations

$$
\sqrt{5-x}=0, \quad 3^{x^{2}-7.2 x+3.9}=9 \sqrt{3}
$$

From the first equation, we find $x_{1}=5$.

To solve the second equation, we transform its right-hand side: $9 \sqrt{3}=3^{2} 3^{1 / 2}=3^{2.5}$. Thus, the second equation of the system is equivalent to the equation

$$
x^{2}-7.2 x+3.9=2.5
$$

i.e.,

$$
x^{2}-7.2 x+1.4=0
$$

From this, we find $x_{2}=1 / 5, x_{3}=7$.

## 82  CH. 2. EQUATIONS WITH ONE UNKNOWN

The domain of permissible values of the equation includes only the number 1/5. Therefore, the solutions of equation (1) are the numbers 5 and $1 / 5$.

Example 7. Solve the equation

$$
3^{2 x-1} 5^{3 x+2}=(9 / 5) 5^{2 x} 3^{3 x}
$$

Solution. In this equation, there are two different bases of powers. Dividing both sides of equation (2) by the positive quantity ( $9 / 5$ ) $5^{2 x} 3^{3 x}$, we obtain the equation

$$
\frac{3^{2 x-1} 5^{3 x+2}}{(9 / 5) 5^{2 x} 3^{3 x}}=1 \Leftrightarrow 5^{x+3} \cdot 3^{-x-3}=1
$$

The last equation is equivalent to the equation

$$
(5 / 3)^{x+3}=(5 / 3)^{0} ;
$$

from this, $x+3=0$, i.e., $x=-3$.