Find the value of $a^{2}+b^{2}+c^{2}-2ab+2ac-2bc$,if $a=x+y$,$b=x-y$,and $c=2x-1$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} = 49$
$II.$ $y^{2} + 15y + 56 = 0$

The expression $x^{4}+7 x^{2}+16$ can be factored as:

Consider the equation $x^2 + \alpha x + \beta = 0$ having roots $\alpha, \beta$ such that $\alpha \neq \beta$. Also consider the inequality $| |y - \beta| - \alpha | < \alpha$,then:

Solve the given two equations and select the correct answer from the given options.
$I.$ $81x^2 - 9x - 2 = 0$
$II.$ $56y^2 - 13y - 3 = 0$

The two roots of the equation $x^3 - 9x^2 + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots are: