Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-11x+24=0$
$II.$ $2y^{2}-9y+9=0$

If $\alpha$ and $\beta$ are the roots of the equation $a x^{2}+b x+c=0$,find the value of $\alpha^{2}+\beta^{2}$.

The reciprocal of $x+\frac{1}{x}$ is

If $ax^2 + bx + c = 0$ has real and distinct roots,$\alpha$ and $\beta$ $(\beta > \alpha)$. Further $a > 0, b < 0$ and $c < 0$ then :-

$Quantity \, 1$: Value of $y$. $A$ vessel contains $2.5 \, \text{liters}$ of water and $10 \, \text{liters}$ of milk. $20 \%$ of the contents of the vessel are removed. To the remaining contents,$x$ liters of water are added to reverse the ratio of water and milk. Then $y$ liters of milk are added again to reverse the ratio of water and milk. $Quantity \, 2$: $120 \, \text{liters}$.

One root of the quadratic equation $x^{2}-5x+6=0$ is $3$. Find the other root.