If the value of the discriminant of the quadratic equation $x^{2} + \sqrt{2}x + 2 = 0$ is .... .

What is the value of $k$ if one of the roots of the quadratic equation $x^{2} + 6x + k = 0$ is $-4$?

Find whether the following equation has real roots. If real roots exist,find them.
$\frac{1}{2x-3} + \frac{1}{x-5} = 1, x \neq \frac{3}{2}, 5$

Obtain the roots of the following quadratic equation using the quadratic formula: $\sqrt{3} x^{2} + 10 x - 8 \sqrt{3} = 0$.

Find the roots of the following quadratic equation using the quadratic formula,if they exist: $\frac{1}{x+1} + \frac{2}{x+2} = \frac{4}{x+4}$; $(x \neq -1, -2, -4)$

Solve the following equation using the method of factorization and write its solution set: $\frac{x}{1-x} + \frac{1-x}{x} = \frac{13}{6}$