Sum of roots

Let $f(x) = |x^2 - 9| + |x^2 - 5| - |\lambda - 5|$ , then for $f(x) = 0$ will get all possible solutions Then, $f(x) = \left\{\begin{matrix} 2x^2 - 14 - |\lambda - 5|, |x| > 3\\ 4 - |\lambda -5| , \sqrt{5} \leqslant |x| \leqslant 3 \\ 14 - 2x^2 - |\lambda -5|, |x| < \sqrt{5} \end{matrix}\right.$

Now, there will be infinite solutions of $x$ only if f(x) = 0 (constant) in some interval. Thus, when $\sqrt{5} \leqslant |x| \leqslant 3$ , $f(x)$ behaves as a constant function. So, putting $f(x) = 0$ , we get $4 - |\lambda -5| = 0$

$\Rightarrow$ $|\lambda - 5| = 4 \Rightarrow \lambda = 5 ± 4$ So, $\lambda_1 = 9$ and $\lambda_2 = 1$

Thus $\lambda_1 + \lambda_2 = 9 + 1 = 10$ :- Answer