If $x=-2$ is a solution of a quadratic equation $k x^{2}+5 x+2=0,$ then $k=\ldots \ldots \ldots \ldots$

The denominator of a non-zero ratio is one less than twice the numerator. If the sum of the ratio and its reciprocal is $2 \frac{4}{15},$ find the ratio.

Find the discriminant of the following quadratic equation: $5x^{2} + 2x - 1 = 0$.

Solve the following equation using the method of factorization and write its solution set: $x + 2 - \frac{6}{x + 2} = 1$

Find a natural number whose square diminished by $84$ is equal to thrice of $8$ more than the given number.

Verify whether the given value of $x$ is a solution of the quadratic equation or not: $4 \sqrt{3} x^{2} + 5 x - 2 \sqrt{3} = 0$; $x = \frac{\sqrt{3}}{4}$.