The expression $x^2 + 2bx + c$ has a positive value for all real $x$ if:

Solve the given two equations and select the correct answer from the given options.
$I.$ $\frac{9}{\sqrt{x}} + \frac{19}{\sqrt{x}} = \sqrt{x}$
$II.$ $y^{5} - \frac{(2 \times 14)^{11/2}}{\sqrt{y}} = 0$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+12x+36=0$
$II.$ $y^{2}+15y+56=0$

If $x = 2^{1/3} - 2^{-1/3}$,find the value of $2x^3 + 6x$.

If $a^{2}=by+cz, b^{2}=cz+ax, c^{2}=ax+by,$ then the value of $\frac{x}{a+x}+\frac{y}{b+y}+\frac{z}{c+z}$ is

If the equations $2x^2 + 3x + 5\lambda = 0$ and $x^2 + 2x + 3\lambda = 0$ have a common root,then $\lambda = $