Guaranteed Real Roots

Algebra Level 5

Find the largest positive integer $n$ such that for all real numbers $a_1$ , $a_2$ , $\dots , a_{n+1}$ , the equation $a_{n+1} x^2 - 2x \sqrt{ a_1^2 + a_2^2 + \cdots + a_{n+1}^2} + (a_1 + a_2 + \cdots + a_n) = 0$ has real roots.

Details and assumptions

Clarification: $a_1, a_2, \ldots, a_{n+1}$ is any set of real numbers. There is no restrictions stated in the problem. They do not need to be integers, nor always positive, nor an arithmetic progression.