Learn Russian: Равновесие зависит от обоих решений.

The noun решение (decision) is neuter. The word for both has two base forms: оба for masculine and neuter nouns, and обе for feminine nouns. Since we need the neuter form in the genitive case, it becomes обоих. For a feminine noun, you would use обеих.

I learned that оба is followed by the genitive singular. Why is решений in the genitive plural?

This is a very common trap! In the nominative case and inanimate accusative, оба acts like the number two and is followed by the genitive singular: оба решения. But in all other cases, like the genitive case used here, оба acts like an adjective, and the noun matches it in the plural. So, both the quantifier and the noun take the genitive plural: обоих решений.

Why are we using the genitive case at all?

Because of the phrase depends on. In Russian, the verb зависеть (to depend) pairs with the preposition от (from), and the preposition от always requires the genitive case.

How is the genitive plural решений formed?