Quadratic Equations#0

Consider equation 1:

$\begin{aligned} (2015x)^2 - (2014)(2016)x - 1 & = 0 & \small \color{#3D99F6} \text{Let }n = 2015 \\ n^2x^2 - (n-1)(n+1)x - 1 & = 0 \\ n^2x^2 - (n^2-1)x - 1 & = 0 \\ (n^2x+1)(x-1) & = 0 \end{aligned}$

$\implies x = \begin{cases} - \dfrac 1{n^2} \\ 1 \end{cases} \implies \alpha = 1$

Consider equation 2:

$\begin{aligned} x^2 +2014x - 2015 & = 0 \\ x^2 + (n-1)x-a & = 0 \\ (x+n)(x-1) & = 0 \end{aligned}$

$\implies x = \begin{cases} -n \\ 1 \end{cases} \implies \beta = -n = - 2015$

Therefore, $\sqrt {\dfrac {\alpha - \beta}{224}} = \sqrt {\dfrac {1 - 2015}{224}} = \sqrt 9 = \boxed{3}$ .