Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+5x-6=0$
$II.$ $2y^{2}-11y+15=0$

If ${x_1}, {x_2}, {x_3}$ are distinct roots of the equation $ax^2 + bx + c = 0$,then:

If the roots of the equation $ax^2 + bx + c = 0$ are reciprocal to each other,then:

Number of natural solutions of the equation $xyz = 2^5 \times 3^2 \times 5^2$ is equal to

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:

Determine $k$ such that the quadratic equation $x^{2}-2(1+3k)x+7(3+2k)=0$ has equal roots.