Let the mean and the variance of 20 observations $x_{1}, x_{2}, \ldots, x_{20}$ be 15 and 9 , respectively. For $\alpha \in \mathbf{R}$, if the mean of $$\left(x_{1}+\alpha\right)^{2},\left(x_{2}+\alpha\right)^{2}, \ldots,\left(x_{20}+\alpha\right)^{2}$$ is 178 , then the square of the maximum value of $\alpha$ is equal to ________.

<p>Given $$\sum\limits_{{{i = 1} \over {20}}}^{20} {{x_i} = 15 \Rightarrow \sum\limits_{i = 1}^{20} {{x_i} = 300} } $$</p> <p>and $$\sum\limits_{{{i = 1} \over {20}}}^{20} {x_i^2 - {{\left( {\overline x } \right)}^2} = 9 \Rightarrow \sum\limits_{i = 1}^{20} {x_i^2 = 4680} } $$</p> <p>Mean $$ = {{{{({x_i} + \alpha )}^2} + {{({x_2} + \alpha )}^2}\, + \,.....\, + \,{{({x_{20}} + \alpha )}^2}} \over {20}} = 178$$</p> <p>$$ \Rightarrow {{\sum\limits_{i = 1}^{20} {x_i^2 + 2\alpha \sum\limits_{i = 1}^{20} {{x_i} + 20{\alpha ^2}} } } \over {20}} = 178$$</p> <p>$\Rightarrow 4680 + 600\alpha + 20{\alpha ^2} = 3560$</p> <p>$\Rightarrow {\alpha ^2} + 30\alpha + 56 = 0$</p> <p>$\Rightarrow {\alpha ^2} + 28\alpha + 2\alpha + 56 = 0$</p> <p>$\Rightarrow (\alpha + 28)(\alpha + 2) = 0$</p> <p>${\alpha _{\max }} = - 2 \Rightarrow \alpha _{\max }^2 = 4.$</p>