Every French noun carries a grammatical gender and a number, and adjectives have to agree with both — masculine nouns pair with an adjective's base form, feminine nouns usually add an -e, and either way, a plural noun adds an -s on top of whichever gender form you started with. Most adjectives follow that predictable pattern, but a handful of the most common ones — beau, vieux, nouveau — break it twice over: they change enough between masculine and feminine to be worth learning as sets, including a rarely-taught third form used only right before a masculine noun that starts with a vowel sound, and in the plural, beau and nouveau swap the expected -s for an -x instead.

The short version

Masculine nouns take the adjective's base form. Feminine nouns usually add -e, though the exact change depends on how the masculine form ends. Either way, a plural noun then adds an -s on top of whichever gender form you started with -- or, for a few adjectives like beau and nouveau, an -x instead.

Masculine

Base form of the adjective, +s for the plural

un grand hommeun livre intéressantle vieux quartierles grands hommes
un grand homme
a tall man
Masculine noun, base adjective form
un livre intéressant
an interesting book
Already ends in a consonant sound -- no change needed for masculine
un beau jardin
a beautiful garden
Irregular masculine form, used before a consonant sound
les grands hommes
the tall men
Plural just adds -s on top -- the masculine form still doesn't take an -e

Feminine

Usually the masculine form + -e, +s for the plural

une grande femmeune histoire intéressantela vieille villeles grandes femmes
une grande femme
a tall woman
Add -e to the masculine form
une histoire intéressante
an interesting story
Same rule: add -e, which also makes the final consonant audible
une belle maison
a beautiful house
Beau's irregular feminine pair, not just beau + e
les grandes femmes
the tall women
Both markers stack -- -e for feminine, then -s for plural

C'est un vieil ami, pas une vieille amie.

He's an old (male) friend, not an old (female) friend.

Beau, nouveau, and vieux each have a special third form -- bel, nouvel, vieil -- used only right before a masculine noun that starts with a vowel sound, purely so the two words flow together. It's still the masculine form (ami stays a male friend), just spelled to avoid a vowel collision; vieille is the separate feminine form, used for amie regardless of what sound follows. The same three adjectives carry a second irregularity into the plural: beau and nouveau take -x instead of -s (de beaux jardins, de nouveaux amis), while vieux, already ending in -x in the singular, just stays vieux (de vieux amis).

Think you've got it?

Take the quiz and see if you can match adjectives to the noun they're describing.