Solve the following equation:

$$
\left(\frac{7}{5}\right)^{2 x+1}-\left(\frac{7}{5}\right)^{x+2}+\frac{24}{35}=0
$$
Let

$$
\left(\frac{7}{5}\right)^{x}=y
$$

then

$$
\frac{7}{5} y^{2}-\frac{49}{25} y+\frac{24}{35}=0
$$

from which

$$
y_{1}=\frac{5}{7}, y_{2}=\frac{24}{35}=0
$$

and thus

$$
\left(\frac{7}{5}\right)^{x_{1}}=\frac{5}{7} \text { and }\left(\frac{7}{5}\right)^{x_{2}}=\frac{24}{35}
$$

from which

$$
x_{1}=-1, x_{2}=-\frac{5462}{4871}
$$

(Pichler Sándor, Budapest, VI. district, high school)

The problem was also solved by: Harsányi Z., Heimlich P., Jánosy Gy., Schlesinger Gy., Schwarz Gy., Schuster Gy., Szóke D., Tóth B., Weber Gy.