〜だ (na-adjective copula)

Na-adjectives — also called "adjectival nouns" — are a distinct class from i-adjectives. They behave grammatically like nouns: they take the copula だ / です to form a predicate, and modify nouns with な instead of い (静かな部屋 — "a quiet room").

The naming reflects the modifying form: 静かな, きれいな, 元気な. The plain predicate form is just 静かだ, 元気だ — the な appears only when there's a noun being modified.

Conjugation tracks the copula, not adjective morphology:
• Plain non-past: 静かだ
• Plain past: 静かだった
• Plain negative: 静かじゃない (or 静かではない)
• Plain past negative: 静かじゃなかった
• Polite forms: 静かです, 静かでした, 静かじゃないです (or 静かではありません)

The two-class system (i-adjective vs na-adjective) is one of the first hurdles in Japanese. The traps: きれい looks like an i-adjective because it ends in い, but it's a na-adjective (きれいな, きれいだ). 嫌い also ends in い but is a na-adjective. There's no derivable rule — these are memorized on a per-adjective basis.