Construct a quadratic equation whose roots have the sum $= 6$ and product $= -16$.

Solve the given two equations and select the correct answer from the given options.
$I. x^{2}-2x-15=0$
$II. y^{2}+5y+6=0$

If $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

Solve the given two equations and select the correct option.
$I.$ $12x^2 - 47x + 40 = 0$
$II.$ $4y^2 + 3y - 10 = 0$

If $\alpha, \beta$ are the roots of the equation $x^2 - 2x + 4 = 0$,then the value of $\alpha^n + \beta^n$ is

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then the quadratic equation whose roots are $(\alpha^2 - \beta^2)(\alpha^3 - \beta^3)$ and $\alpha^3\beta^2 + \alpha^2\beta^3$ is (where $S = p[p^4 - 5p^2q + 5q^2]$ and $P = p^2q^2(p^4 - 5p^2q + 4q^2)$).