Solve the given two equations and select the correct answer from the given options.
$I. x - 7\sqrt{3x} + 36 = 0$
$II. y - 12\sqrt{2y} + 70 = 0$

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies:

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

If the equations $2ax^2 - 3bx + 4c = 0$ and $3x^2 - 4x + 5 = 0$ have a common root,then $\left( \frac{a + b}{c} \right)$ is equal to (where $a, b, c \in R$).

$Quantity \, 1$: 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. $Quantity \, 2$: $120 \, \text{liters}$.

If one root of the equation $ax^2 + bx + c = 0$ is the square of the other,then $a(c - b)^3 = cX$,where $X$ is