Solve the equation: $\;$ $5^{2x} - 7^x - 5^{2x} \times 35 + 7^x \times 35 = 0$
Given equation: $\;\;$ $5^{2x} - 7^x - 5^{2x} \times 35 + 7^x \times 35 = 0$
i.e. $\;$ $5^{2x} - 7^x = 5^{2x} \times 35 - 7^x \times 35$
i.e. $\;$ $5^{2x} - 7^x = 35 \left(5^{2x} - 7^x\right)$ $\;\;\; \cdots \; (1)$
In equation $(1)$, if $\;$ $5^{2x} - 7^x \neq 0$, $\;$ then dividing both sides of equation $(1)$ by $5^{2x} - 7^x$ gives
$1 = 35$ $\;\;$ which is not possible.
$\implies$ In equation $(1)$, $\;\;$ $5^{2x} - 7^x = 0$
i.e. $\;$ $\left(5^2\right)^x = 7^x$
i.e. $\;$ $25^x = 7^x$
i.e. $\;$ $\dfrac{25^x}{7^x} = 1$
i.e. $\;$ $\left(\dfrac{25}{7}\right)^x = 1$
i.e. $\;$ $\left(\dfrac{25}{7}\right)^x = \left(\dfrac{25}{7}\right)^0$
$\implies$ $x = 0$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = 0$