The charachteristic polynomial of two similar matrices are the same. Thus, if has the charachteristic polynomial then by the Cayley-Hamilton theorem. Of course, this theorem is very advanced and you probably never seen it before. Therefore, there is a weaker version for this theorem which states that a diagnolizable matrix satisfies its charachteristic polynomial. Since is a symettric matrix it means it is diagnolizable and the rest follows.

Thus, you need to find which solve this polynomial equation.