Solve the given two equations and select the correct option.
$I.$ $x^{2}+5x+6=0$
$II.$ $y^{2}+3y+2=0$

The expression $x^2-x+1$ has:

If $x = \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and $y = \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$,then the value of $\frac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}} = ?$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt{172}$
$II.$ $y^{2} - 29y + 210 = 0$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{3} - 468 = 1729$
$II.$ $y^{2} - 1733 + 1564 = 0$

$Quantity \, 1:$ Original duration of flight. In a flight of $3000 \, km$,an aircraft was slowed down by bad weather. Its average speed for the trip was reduced by $100 \, km/hr$ and the time increased by $1 \, hour$.
$Quantity \, 2:$ Usual time of a man who,when walking at $\frac{3}{4}$ of his usual pace,reaches his office $20 \, minutes$ late.