The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

The coefficient of $x$ in the equation $x^2 + px + q = 0$ was taken as $17$ in place of $13$. Its roots were found to be $-2$ and $-15$. The roots of the original equation are:

Find the complete set of values of $k$ for which the equation $4^x - (k + 2)2^x + 2k = 0$ has exactly one positive root.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{3} - 468 = 1729$
$II.$ $y^{2} - 1733 + 1564 = 0$

Solve the given two equations and select the correct option.
$I.$ $77 x^{2}+58 x+8=0$
$II.$ $42 y^{2}+59 y+20=0$

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to: