Solve the given two equations and select the correct option.
$I.$ $x^{2}-19x+84=0$
$II.$ $y^{2}-25y+156=0$

If ${x^2} + {y^2} = 25$ and $xy = 12$,then find the possible values of $x$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt{625}$
$II.$ $y = \sqrt{676}$

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in $G.P.$,where $\Delta = b^2 - 4ac$,then:

If $\alpha$ and $\beta$ are roots of the equation $Ax^2 + Bx + C = 0$,then the value of $\alpha^3 + \beta^3$ is

If $8, 2$ are the roots of ${x^2} + ax + \beta = 0$ and $3, 3$ are the roots of ${x^2} + \alpha x + b = 0$,then the roots of ${x^2} + ax + b = 0$ are