Phương trình bậc ba và một số vấn đề liên quan trong hình học phẳng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

❇Ị■ ❚➷◆ ◆Ú ❚❍❆◆❍ ❳❯❹◆

P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆
❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆
❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵



❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

❇Ị■ ❚➷◆ ◆Ú ❚❍❆◆❍ ❳❯❹◆

P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆
❱⑨ ▼❐❚ ❙➮ ❱❻◆ ✣➋ ▲■➊◆ ◗❯❆◆
❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆●
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆

Pữỡ t sỡ
số

ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿

❚❙✳ ▲➊ ❚❍❆◆❍ ❍■➌❯

❇➻♥❤ ✣à♥❤ ✲ ✷✵✷✵



ử ử
é
ởt số ỵ
❇➚

✐✐✐
✈
✶

✶✳✶

❈⑩❈ ❇❻❚ ✣➃◆● ❚❍Ù❈ ✣❸■ ❙➮ ◗❯❆◆ ❚❘➴◆●

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶

✶✳✷

❈⑩❈ ❈➷◆● ❚❍Ù❈ ▲×Đ◆● ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷

✶✳✸

❈⑩❈ ❍➏ ❚❍Ù❈ ▲×Đ◆● ❈❒ ❇❷◆ ❚❘❖◆● ❚❆▼ ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✹

❚➑◆❍ ❈❍❻❚ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆

✻

✶✳✺

▼➮■ ▲■➊◆ ❍➏ ●■Ú❆ ❈⑩❈ ◆●❍■➏▼ ❈Õ❆ ▼❐❚ ❙➮ P❍×❒◆● ❚❘➐◆❍ ❇❾❈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❇❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✷ P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆ ❱⑨ ❍➏ ❚❍Ù❈ ▲×Đ◆● ❚❘❖◆● ❚❆▼ ●■⑩❈ ✶✾
✷✳✶

✷✳✷

P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆ ❱⑨ ❈⑩❈ ❍➏ ❚❍Ù❈ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❈⑩❈
✣×❮◆● ❈Õ❆ ❚❆▼ ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✶✳✶

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❝↕♥❤ t❛♠ ❣✐→❝ ✳ ✳ ✳

✶✾

✷✳✶✳✷

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ✤÷í♥❣ ❝❛♦ t❛♠ ❣✐→❝

✷✷

✷✳✶✳✸

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ✤÷í♥❣ tr✉♥❣ t✉②➳♥

✷✹

✷✳✶✳✹

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝

✳

✷✻

✷✳✶✳✺

❍➺ t❤è♥❣ ❜➔✐ t➟♣ ❝→❝ ❤➺ t❤ù❝ ✈➲ ❝→❝ ✤÷í♥❣ ❝õ❛ t❛♠ ❣✐→❝

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆ ❱⑨ ❈⑩❈ ❍➏ ❚❍Ù❈ ▲■➊◆ ◗❯❆◆ ✣➌◆ ❇⑩◆
❑➑◆❍ ❈⑩❈ ✣×❮◆● ❚❘➪◆ ◆❐■✱ ◆●❖❸■ ❱⑨ ❇⑨◆● ❚■➌P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥
❜➔♥❣ t✐➳♣ t❛♠ ❣✐→❝

✷✳✷✳✷

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❜→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣
trá♥ ♥ë✐✱ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷✳✸

✸✽

✹✵

❍➺ t❤è♥❣ ❜➔✐ t➟♣ ❝→❝ ❤➺ t❤ù❝ ✈➲ ❜→♥ ❦➼♥❤ ❝→❝ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣✱
♥ë✐✱ ♥❣♦↕✐ t✐➳♣ ❝õ❛ t❛♠ ❣✐→❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✐

✹✶


✷✳✸

P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆ ❱⑨ ❈⑩❈ ❍➏ ❚❍Ù❈ ❱➋ ❈⑩❈ ●➶❈ ❈Õ❆ ❚❆▼
●■⑩❈

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✷✳✸✳✶

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❤➔♠ s✐♥ ❝→❝ ❣â❝ ✳ ✳

✹✼

✷✳✸✳✷

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❤➔♠ ❝æs✐♥ ❝→❝ ❣â❝

✹✾

✷✳✸✳✸

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❝â ♥❣❤✐➺♠ ❧➔ ❝→❝ ②➳✉ tè t❤❡♦ ❤➔♠ t❛♥❣ ❤♦➦❝
❝æt❛♥❣ ❝→❝ ❣â❝

✷✳✸✳✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✷

❍➺ t❤è♥❣ ❜➔✐ t➟♣ ♠ët sè ❤➺ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ tr♦♥❣ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳

✺✺

✸ ❈⑩❈ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❚❘❖◆● ❚❆▼ ●■⑩❈
✸✳✶

✸✳✷

✻✼

❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❈⑩❈ ✣×❮◆● ❈Õ❆ ❚❆▼ ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻✼

✸✳✶✳✶

❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ❝→❝ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝

✻✼

✸✳✶✳✷

❇➜t ✤➥♥❣ t❤ù❝ ❤➻♥❤ ❤å❝ ❝❤ù❛

✳ ✳ ✳ ✳ ✳

✽✶

✸✳✶✳✸

❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ❝→❝ ✤÷í♥❣ ❝❛♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✹

✸✳✶✳✹

❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ✤÷í♥❣ tr✉♥❣ t✉②➳♥

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽✼

✸✳✶✳✺

❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ❝→❝ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✵

✸✳✶✳✻

❇➜t ✤➥♥❣ t❤ù❝ ✈➲ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

x = p − a, y = p − b, z = p − c

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✶

❇❻❚ ✣➃◆● ❚❍Ù❈ ❱➋ ❈⑩❈ ●➶❈ ❈Õ❆ ❚❆▼ ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✺

sin A, sin B, sin C ✳
cos A, cos B, cos C

✸✳✷✳✶

❇➜t ✤➥♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝❤ù❛

✸✳✷✳✷

❇➜t ✤➥♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝❤ù❛

✸✳✷✳✸

❇➜t ✤➥♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝❤ù❛ t❛♥❣ ❤♦➦❝ ❝ỉt❛♥❣ ❝→❝ ❣â❝

B
C
A
sin , sin , sin
2
2
2
A
B
C
cos , cos , cos
2
2
2

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✵

✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✸

✸✳✷✳✹

❇➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛

✸✳✷✳✺

❇➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶✼

✸✳✷✳✻

❇➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ t❛♥❣ ❤♦➦❝ ❝æt❛♥❣ ❝→❝ ♥û❛ ❣â❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷✵

✸✳✷✳✼

❇➜t ✤➥♥❣ t❤ù❝ ❝❤ù❛ ❜➻♥❤ ♣❤÷ì♥❣ ❝õ❛ t❛♥❣ ❤❛② ❝ỉt❛♥❣ ❝→❝ ♥û❛ ❣â❝

✶✷✸

❑➌❚ ▲❯❾◆
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳

✶✷✻
✶✷✼

✐✐


▲❮■ ▼Ð ✣❺❯
✣↕✐ sè✱ ●✐↔✐ t➼❝❤ ✈➔ ❍➻♥❤ ❤å❝ ❧➔ ♥❤ú♥❣ ♣❤➙♥ ♠ỉ♥ ❝ì ❜↔♥ ❝➜✉ t❤➔♥❤ ♥➯♥ ♠ỉ♥ ❚♦→♥ ð ♣❤ê
t❤ỉ♥❣✳ ❈❤ó♥❣ ❝â ♠è✐ ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ✈ỵ✐ ♥❤❛✉ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ♥➔② ❧➔ ♠ët ✈➜♥ ✤➲ ✤→♥❣ q✉❛♥
t➙♠ ❤✐➺♥ ♥❛②✳ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ tr♦♥❣ ✤↕✐ sè ✈➔ ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ t❛♠ ❣✐→❝
tr♦♥❣ ❍➻♥❤ ❤å❝ ♣❤➥♥❣✱ ✤➲✉ ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❣➛♥ ❣ơ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ❛✐ ❤å❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉
❚♦→♥ ❤å❝✳ ❈❤ó♥❣ ❝ơ♥❣ t❤÷í♥❣ ❤❛② ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ❝➜♣✳
❚✉② ♥❤✐➯♥✱ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝❤ó♥❣✱ ✤➦❝ ❜✐➺t ❧➔ ♣❤÷ì♥❣ ♣❤→♣ sû ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✤➸
❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✱ ✤÷í♥❣ trá♥ ❧↕✐ ❝❤÷❛ ✤÷đ❝ ♥❤✐➲✉ ♥❣÷í✐ ❦❤→♠ ♣❤→✳
❇↔♥ t❤➙♥ t→❝ ❣✐↔ ❝❤♦ r➡♥❣ ✤➙② ❧➔ ♠ët ❝❤✉②➯♥ ✤➲ tữỡ ố õ ố ợ ồ s ờ
tổ ✈➔ ❝ơ♥❣ ♣❤ị ❤đ♣ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ❝➜♣✳
▲✉➟♥ ✈➠♥ ✏

♣❤➥♥❣ ✑

P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✈➔ ♠ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ tr♦♥❣ ❤➻♥❤ ❤å❝

♥❤➡♠ ♠ö❝ ✤➼❝❤ t➻♠ ❤✐➸✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✱ ✤➦❝ ❜✐➺t ❧➔ t➼♥❤

❝❤➜t ❝õ❛ ❝→❝ ♥❣❤✐➺♠✱ t➻♠ tá✐ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ♠ët ✤è✐ t÷đ♥❣ ✤↕✐ sè ❧➔ ữỡ tr
ợ ồ ự ử ❝õ❛ ♥â ✈➔♦ ✈✐➺❝ ❣✐↔✐ ♠ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❤➺
t❤ù❝ ❧÷đ♥❣ tr♦♥❣ t❛♠ ❣✐→❝ ✈➔ ✤÷í♥❣ trá♥✳ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠
❦❤↔♦✱ ỗ õ ữỡ ữ s

ữỡ t❤ù❝ ❝❤✉➞♥ ❜à

✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ♣❤ö❝

✈ö ❝❤♦ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ð ♣❤➛♥ s ở ỗ t tự ỡ ❝→❝ ❤➺ t❤ù❝
❧÷đ♥❣ ❝ì ❜↔♥ tr♦♥❣ t❛♠ ❣✐→❝ ✈➔ ✤÷í♥❣ trá♥✱✳✳✳❀ ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ❝ỉ♥❣ t❤ù❝ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥
✤➳♥ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ♥❤÷ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠✱ ✣à♥❤ ❧➼ ❱✐➧t❡✱ ❝→❝ t➼♥❤ ❝❤➜t ✤è✐
①ù♥❣✱ ✣à♥❤ ❧➼ ❙t✉r♠✱ sü ①→❝ ✤à♥❤ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ✸ ♥❤➟♥ ❜❛ sè ✤➦❝ ❜✐➺t ❧➔♠ ♥❣❤✐➺♠✱ ✳ ✳ ✳

❈❤÷ì♥❣ ✷✿ P❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✈➔ ❤➺ t❤ù❝ ❧÷đ♥❣ tr♦♥❣ t❛♠ ❣✐→❝

tr➻♥❤ ❜➔② ♠ët sè ❞↕♥❣ t♦→♥ ✈➲ ❝→❝

✳ ❈❤÷ì♥❣ ♥➔②

✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤÷í♥❣ ✈➔ ❝→❝ ❣â❝ ❝õ❛ t❛♠

❣✐→❝ ❜➡♥❣ ❝→❝❤ →♣ ❞ư♥❣ ❝→❝ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr ở ừ
ỗ tự ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ ✈➲ ❝→❝ ❝↕♥❤✱ ❝→❝ ✤÷í♥❣
❝❛♦✱ tr✉♥❣ t✉②➳♥✱ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝✱ ❜→♥ ❦➼♥❤ ❝õ❛ ❝→❝ ✤÷í♥❣ trá♥ ♥ë✐✱ ♥❣♦↕✐ ✈➔ ❜➔♥❣ t✐➳♣ ✈➔
❝→❝ ❣â❝ ❝õ❛ t❛♠ ❣✐→❝✳

❈❤÷ì♥❣ ✸✿ ❈→❝ ❜➜t tự tr t

ữỡ trữợ ữỡ tr ❜➔② ❤➺ t❤è♥❣ ❝→❝

✐✐✐

✳ ❉ü❛ ✈➔♦ ❝→❝ ❦➳t q✉↔ ✤➣ ❜✐➳t tr♦♥❣

❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✳


rữợ tr ở ừ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤
✈➔ s➙✉ s tợ ữớ t t ữợ õ t t
♥➔②✳ ❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t♦➔♥ t❤➸ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣
❦❤♦❛ ❚♦→♥ ✈➔ t❤è♥❣ ❦➯✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ◗✉② ◆❤ì♥ ✤➣ ❞↕② ❜↔♦ ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝
t➟♣ t↕✐ ❚r÷í♥❣ ✤➸ ❡♠ ❝â ✤÷đ❝ ♥➲♥ t↔♥❣ tr✐ t❤ù❝ ❝ơ♥❣ ♥❤÷ ❦✐♥❤ ♥❣❤✐➺♠ ở số qỵ
tr tữỡ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤
❧➺ ✈➔ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ò♥❣✱ ❡♠ ỷ
ớ ỡ t tợ ỵ t ổ tr ở ỗ tớ qỵ
t õ ỵ ỳ ❝á♥ t❤✐➳✉ sât✱ ❣✐ó♣ ❡♠ rót r❛ ✤÷đ❝ ❦✐♥❤ ♥❣❤✐➺♠
❝❤♦ ❧✉➟♥ ✈➠♥ ❝ơ♥❣ ♥❤÷ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ s❛✉ ♥➔②✳ t ữủ sỹ t
t ừ ỵ t ổ ụ ữ sỹ õ ỵ ừ
❞ị ❤➳t sù❝ ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣
♥❤➟♥ ✤÷đ❝ sü õ ỵ ỳ ỵ ừ ỵ t❤➛② ❝ỉ ✈➔ ❜↕♥ ✤å❝✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤
❝↔♠ ì♥✦
◗✉② ◆❤ì♥✱ ♥❣➔② ✷✽ t❤→♥❣ ✼ ♥➠♠ ✷✵✷✵
❍å❝ ✈✐➯♥

❇ị✐ ❚ỉ♥ ◆ú ❚❤❛♥❤ ❳✉➙♥

✐✈


ởt số ỵ
a, b, c
a+b+c
p=

2
S, SABC
r, R
ra :
ha :
ma :
la :

✤ë ❞➔✐ ❝→❝ ❝↕♥❤

BC, CA, AB

❝õ❛ t❛♠ ❣✐→❝ABC.

♥û❛ ❝❤✉ ✈✐ ❝õ❛ t❛♠ ❣✐→❝✳
❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝

ABC.

❧➛♥ ❧÷đt ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐✱ ♥❣♦↕✐ t✐➳♣ ❝õ❛ t❛♠ ❣✐→❝✳

A ❝õ❛
ABC.

❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ ù♥❣ ✤➾♥❤
✤÷í♥❣ ❝❛♦ ♥è✐ tø ✤➾♥❤

A

❝õ❛ t❛♠ ❣✐→❝

✤÷í♥❣ tr✉♥❣ t✉②➳♥ ♥è✐ tø ✤➾♥❤
✤÷í♥❣ ♣❤➙♥ ❣✐→❝ ♥è✐ tø ✤➾♥❤

✈

A

A

ABC.
ABC.

❝õ❛ t❛♠ ❣✐→❝

❝õ❛ t❛♠ ❣✐→❝

t❛♠ ❣✐→❝

ABC.



❈❤÷ì♥❣ ✶
❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
✶✳✶ ❈⑩❈ ❇❻❚ ✣➃◆● ❚❍Ù❈ ✣❸■ ❙➮ ◗❯❆◆ ❚❘➴◆●
❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②
❆▼✲●▼✮✳

✭❇➜t ✤➥♥❣ t❤ù❝ tr✉♥❣ ❜➻♥❤ ❝ë♥❣✲ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✱ ❜➜t ✤➥♥❣ t❤ù❝

●✐↔ sû a1, a2, ..., an ❧➔ ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â
a1 + a2 + ... + an
n

√
n
a1 a2 ...an

✭✶✳✶✳✶✮

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a1 = a2 = ... = an.
❇➜t ✤➥♥❣ t❤ù❝ ❇✉♥②❛❦♦✈s❦②✳ ❱ỵ✐ ♠å✐ ❜ë 2n sè t❤ü❝ ai, bi, i = 1, ...n t❛ ❝â
(a1 b1 + a2 b2 + ...an bb )2 ≤ (a21 + a22 + ... + a2n )(b21 + b22 + ... + b2n ).

✭✶✳✶✳✷✮

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tỗ t số t R s bi = tai ❤♦➦❝ ai = tbi, i = 1, 2, ...n.
❇➜t ✤➥♥❣ t❤ù❝ ❈❤❡❜②s❤❡✈✳ ❛✮ ❈❤♦ ❤❛✐ ❞➣② ✤ì♥ ✤✐➺✉ ❝ị♥❣ ❝❤✐➲✉ a1 a2 ... an ✈➔
b1 b2 ... bn . ❑❤✐ ✤â
(a1 + a2 + ... + an )(b1 + b2 + ... + bn ) ≤ n(a1 b1 + a2 b2 + ... + an bn ).
❜✮

❈❤♦ ❤❛✐ ❞➣② ✤ì♥ ✤✐➺✉ tr→✐ ❝❤✐➲✉ a1

a2

(a1 + a2 + ... + an )(b1 + b2 + ... + bn )

...

an

✈➔ b1

b2

...

n(a1 b1 + a2 b2 + ... + an bn ).

✭✶✳✶✳✸✮

bn .

❑❤✐ ✤â
✭✶✳✶✳✹✮

❚r♦♥❣ ❝↔ ❤❛✐ ❞↕♥❣ tr➯♥✱ ❞➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a1 = a2 = ... = an ❤♦➦❝ b1 = b2 =
... = bn .

❇➜t ✤➥♥❣ t❤ù❝ ◆❡s❜✐tt ❝❤♦ ❜❛ sè ❞÷ì♥❣ ●✐↔ sû a , a , ..., a ❧➔ sè t❤ü❝ ❦❤æ♥❣ ➙♠✳ ❑❤✐
✤â

✳

b
c
a

+
+
b+c b+c a+b
✶

1

2

3
.
2

n

✭✶✳✶✳✺✮


❇➜t ✤➥♥❣ t❤ù❝ ❙❝❤✇❛r③ ❈❤♦ a , ..., a ❧➔ ❝→❝ sè t❤ü❝✱ b , ..., b ❧➔ ❝→❝ sè ❞÷ì♥❣✱x ∈ N .
✳

❑❤✐ ✤â

1

n

1

n

a2
(a1 + a2 + ... + an )2
a21 a22
+
+ ... + n =
.
b1
b2
b2
b1 + b2 + ... + bn

∗

✭✶✳✶✳✻✮

❉➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a1 = a2 = ... = an.
❇➜t ✤➥♥❣ t❤ù❝ ❇❡r♥♦✉❧❧✐✳ ❱ỵ✐ x > −1✱ t❛ ❧✉ỉ♥ ❝â
α
✐✮ (1 + x) ≤ 1 + αx ✈ỵ✐ 0 < α < 1;
α
✐✐✮ (1 + x)
1 + αx ✈ỵ✐ α < 0 ❤♦➦❝ 1.

ìẹ

ỳ õ ữủ ❣✐→❝ s❛♦ ❝❤♦ ❝→❝ ❝æ♥❣ t❤ù❝ ❝â ♥❣❤➽❛✳ ❚❛ ❝â

❈→❝ ❤➡♥❣ ✤➥♥❣ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝ì ❜↔♥

✿

a) sin2 α + cos2 α = 1;
sin α
;
b) tan α =
cos α
c) tan α cot α = 1;
1
d) 2 = 1 + tan2 α;
cos α
1
e) 2 = 1 + cot2 α.
sin α

❈æ♥❣ t❤ù❝ ❝ë♥❣ ❣â❝

✭✶✳✷✳✶❛✮
✭✶✳✷✳✶❜✮
✭✶✳✷✳✶❝✮
✭✶✳✷✳✶❞✮
✭✶✳✷✳✶❡✮

✿

a) sin(α ± β) = sin α cos α ± cos α sin α.

✭✶✳✷✳✷❛✮

b) cos(α ± β) = cos α cos β ∓ sin β cos α.
tan α ± tan β
.
c) tan(α ± β) =
1 ∓ tan α tan β

✭✶✳✷✳✷❜✮
✭✶✳✷✳✷❝✮

❈æ♥❣ t❤ù❝ ❣â❝ ♥❤➙♥ ✤æ✐

✿

a) sin 2α = 2 sin α cos α.

✭✶✳✷✳✸✮

b) cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α.
2 tan α
c) tan 2α =
.
1 − tan2 α

✭✶✳✷✳✹✮

❈æ♥❣ t❤ù❝ ❤↕ ❜➟❝

✭✶✳✷✳✺✮

✿

1 − cos 2α
;
2
1 + cos 2α
b) cos2 =
;
2
1 − cos 2α
c) tan2 α =
.
1 + cos 2α
a) sin2 α =

✷

✭✶✳✷✳✻❛✮
✭✶✳✷✳✻❜✮
✭✶✳✷✳✻❝✮


❈æ♥❣ t❤ù❝ ♥❤➙♥ ❜❛

✿

13a) sin 3α = −4 sin3 α + 3 sin α.

✭✶✳✷✳✼❛✮

13b) cos 3α = 4 cos3 α − 3 cos α.

✭✶✳✷✳✼❜✮

▲✐➯♥ ❤➺ ❣✐ú❛ sin 2α, cos 2α ✈➔ tan α

✿

2 tan α
.
1 + tan2 α
1 − tan2 α
b) cos 2α =
.
1 + tan2 α
a) sin 2α =

❈æ♥❣ t❤ù❝ ❜✐➳♥ ✤ê✐ t➼❝❤ t❤➔♥❤ tê♥❣

✭✶✳✷✳✽❛✮

✭✶✳✷✳✽❜✮

✿

1
a) cos α cos β = [cos(α − β) + cos(α + β)].
2
1
b) sin α sin β = [cos(α − β) − cos(α + β)].
2
1
c) sin α cos β = [sin(α − β) + sin(α + β)].
2

✭✶✳✷✳✾❛✮
✭✶✳✷✳✾❜✮
✭✶✳✷✳✾❝✮

❈æ♥❣ t❤ù❝ ❜✐➳♥ ✤ê✐ tê♥❣ t❤➔♥❤ t➼❝❤

✿

α+β
α−β
cos
.
2
2
α+β
α−β
b) sin α − sin β = 2 cos
sin
.
2

2
α−β
α+β
cos
.
c) cos α + cos β = 2 cos
2
2
α+β
α+β
d) cos α − cos β = −2 sin
sin
.
2
2
sin(α ± β)
e) tan α ± tan β =
.
cos α cos β
a) sin α + sin β = 2 sin

✸

✭✶✳✷✳✶✵❛✮
✭✶✳✷✳✶✵❜✮
✭✶✳✷✳✶✵❝✮
✭✶✳✷✳✶✵❞✮
✭✶✳✷✳✶✵❡✮

❈→❝ ❤➺ t❤ù❝ ❧÷đ♥❣ ❣✐→❝ ❝ì ❜↔♥ tr♦♥❣ t❛♠ ❣✐→❝

✿

B
C
A
cos cos .
2
2
2
b) sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
a) sin A + sin B + sin C = 4 cos

✭✶✳✷✳✶✶❛✮
✭✶✳✷✳✶✶❜✮

c) sin2 A + sin2 B + sin2 C = 2(1 + cos A cos B cos C).
A
B
C
d) cos A + cos B + cos C = 1 + 4 sin sin sin .
2
2
2
e) cos 2A + cos 2B + cos 2C = −1 − 4 cos A cos B cos C.

✭✶✳✷✳✶✶❞✮

f ) cos2 A + cos2 B + cos2 C = 1 − 2 cos A cos B cos C.

✭✶✳✷✳✶✶❢ ✮

g)❱ỵ✐

♠å✐ t❛♠ ❣✐→❝

ABC

✭✶✳✷✳✶✶❝✮

✭✶✳✷✳✶✶❡✮

❦❤æ♥❣ ✈✉æ♥❣✱ t❛ ❝â

tan A + tan B + tan C = tan A tan B tan C.
A
B
C
A
B
C
h) cot + cot + cot = cot cot cot .
2
2
2
2
2
2
A

B
B
C
C
A
k) tan tan + tan tan + tan tan = 1.
2
2
2
2
2
2
l) cot A cot B + cot B cot C + cot C cot A = 1.

✭✶✳✷✳✶✶❣✮
✭✶✳✷✳✶✶❤✮
✭✶✳✷✳✶✶✐✮
✭✶✳✷✳✶✶❥✮

m) tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C.

✭✶✳✷✳✶✶❦✮

✶✳✸ ❈⑩❈ ❍➏ ❚❍Ù❈ ▲×Đ◆● ❈❒ ❇❷◆ ❚❘❖◆● ❚❆▼
●■⑩❈
▼ư❝ ♥➔② tâ♠ t➢t ♠ët sè ❤➺ t❤ù❝ ❧÷đ♥❣ ❝ì ❜↔♥ tr♦♥❣ t❛♠ ❣✐→❝ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔
s❛✉ ♥➔②✳ ❚ø ✤➙② ✈➲ s❛✉ t❛ sû ❞ư♥❣ ❝→❝ ❦➼ ❤✐➺✉ s❛✉ ✤➙②✿

• a, b, c
ã p, S

ữủt ở ❝↕♥❤

✭❤❛②

SABC ✮

BC, CA, AB

❝õ❛ t❛♠ ❣✐→❝

ABC;

❧➛♥ ❧÷đt ❧➔ ♥û❛ ❝❤✉ ✈✐✱ ❞✐➺♥ t➼❝❤ ❝õ❛ t❛♠ ❣✐→❝

ABC, p =

a+b+c
;
2

• r, R ❧➛♥ ❧÷đt ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐✱ ♥❣♦↕✐ t✐➳♣ ❝õ❛ t❛♠ ❣✐→❝ ABC;
• ma , ha , lc , ra ❧➛♥ ❧÷đt ❧➔ ✤ë ❞➔✐ ✤÷í♥❣ tr✉♥❣ t✉②➳♥✱ ✤÷í♥❣ ữớ ữớ
trỏ t ự ợ ❝↕♥❤ BC ✭❤❛② ✤➾♥❤ A✮ ❝õ❛ t❛♠ ❣✐→❝ ABC;
❚r♦♥❣ t❛♠ ❣✐→❝ ABC t❛ ❝â

✣à♥❤ ❧➼ ❤➔♠ sè s✐♥

✿

a
b
c
=
=
= 2R.
sin A
sin B
sin C

✣à♥❤ ❧➼ ❤➔♠ sè ❝♦s✐♥

✭✶✳✸✳✶✮

✿

a2 = b2 + c2 − 2bc cos A; b2 = a2 + c2 − 2ac cos B; c2 = a2 + b2 − 2ac cos C.
✹

✭✶✳✸✳✷✮


❍➺ q✉↔ ❧➔

cos A =

a2 + c 2 − b 2
a2 + b2 − c2
b 2 + c 2 − a2
; cos B =

; cos C =
.
2bc
2ac
2ab

❈ỉ♥❣ t❤ù❝ ✤÷í♥❣ tr✉♥❣ t✉②➳♥
m2a =

✭✶✳✸✳✸✮

✿

2(b2 + c2 ) − a2 2 2(a2 + c2 ) − b2 2 2(a2 + b2 ) − c2
; mb =
; mc =
.
4
4
4

✣à♥❤ ❧➼ ❤➔♠ sè t❛♥❣

✭✶✳✸✳✹✮

✿

A−B
tan
a−b

A−B
C
2
=
= tan
tan .
A+B
a+b
2
2
tan
2

✭✶✳✸✳✺✮

✣à♥❤ ❧➼ ✈➲ ❤➻♥❤ ❝❤✐➳✉

✿

B
C
+ cot
2
2
A
B
c = b. cos A + a. cos B = r cot + cot
2
2

a = b. cos C + c. cos B = r cot

; b = c. cos A + a. cos C = r cot

C
A
+ cot
2
2

;

✭✶✳✸✳✻✮

.

❈→❝ ❝æ♥❣ t❤ù❝ t➼♥❤ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝

✿

1
1
1
S = aha = bhb = chc
2
2
2
1
1
1

= bc sin A = ac sin B = ab sin C
2
2
2
abc
= 2R2 sin A sin B sin C =
4R
= p(p − a)(p − b)(p − c) = pr
= ra (p − a) = rb (p − b) = rc (p − c)
√
arb rc
= rra rb rc =
.
rb + rc

✭✶✳✸✳✼✮

❚ø ✤➙② t❛ ❝â ❝→❝ ❤➺ q✉↔ s❛✉ ✤➙②✿

2 p(p − a)(p − b)(p − c)
,
a
1
1
1
1
1
1
1
=

+ + =
+
+ .
r
ra rb rc
ha hb hc

ha =

❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝
R=

✭✶✳✸✳✾✮

✿

a
b
c
abc
=
=
=
=
2 sin A
2 sin B
2 sin C
4S
4
✺

✭✶✳✸✳✽✮

abc
p(p − a)(p − b)(p − c)

.

✭✶✳✸✳✶✵✮


❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ t❛♠ ❣✐→❝
r = (p − a) tan

✿

A
B
C
S
= (p − b) tan = (p − c) tan = =
2
2
2
p

❇→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ❜➔♥❣ t✐➳♣ t❛♠ ❣✐→❝
ra = p tan

A

S
=
=
2
p−a

❈ỉ♥❣ t❤ù❝ ✤÷í♥❣ ♣❤➙♥ ❣✐→❝

(p − a)(p − b)(p − c)
.
p

✭✶✳✸✳✶✶✮

✿

p(p − b)(p − c)
.
p−a

✭✶✳✸✳✶✷✮

✿

A
2 = 2 bcp(p − a) .
b+c
b+c

2bc cos

la =

✭✶✳✸✳✶✸✮

❚❛ ❝â ♠ët sè ❤➺ q✉↔ ✤ì♥ ❣✐↔♥ s❛✉ ✤➙②✿

a + b + c = 2p;

✭✶✳✸✳✶✹✮

ab + bc + ca = p2 + 4Rr + r2 ;

✭✶✳✸✳✶✺✮

abc = 4pRr.

✭✶✳✸✳✶✻✮

❚❤➟t ✈➟②✱ ✤➥♥❣ t❤ù❝ ✤➛✉ t✐➯♥ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❚ø ❤❛✐ ❝æ♥❣ t❤ù❝ t➼♥❤ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝
tr♦♥❣ ✭✶✳✸✳✼✮✿

S∆ABC

abc
= pr =
4R

t❛ s✉② r❛

abc = 4pRr✳

ABC

▼➦t ❦❤→❝✱ ❝ô♥❣ tø ✭✶✳✸✳✼✮ t❛ ❝â

S 2 = p2 r2 = p(p − a)(p − b)(p − c) ⇔ pr2 = p3 − (a + b + c)p2 + (ab + bc + ca)p − abc
⇔ pr2 = p3 − 2p3 + (ab + bc + ca)p − 4pRr
⇔ ab + bc + ca = p2 + 4Rr + r2 .
❉♦ ✤â

ab + bc + ca = p2 + 4Rr + r2 .

ế Pì


ỵ ✭✣à♥❤ ❧➼ ❱✐➧t❡ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✮✳ P❤÷ì♥❣ tr➻♥❤
x3 + ax2 + bx + c = 0

✭✶✳✹✳✶✮

❝â ❜❛ ♥❣❤✐➺♠ ✭❦➸ ❝↔ ♥❣❤✐➺♠ ♣❤ù❝✮ x1, x2, x3 t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙②✿
T1 = x1 + x2 + x3 = −a;

✭✶✳✹✳✷❛✮

T2 = x1 x2 + x2 x3 + x3 x1 = b;

✭✶✳✹✳✷❜✮

T3 = x1 x2 x3 = −c.

✭✶✳✹✳✷❝✮

✻


✣↔♦ ❧↕✐✱ ♥➳✉ ❜❛ sè ♣❤ù❝ x1, x2, x3 t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t ✭✶✳✹✳✷❛✮✱✭✶✳✹✳✷❜✮✱✭✶✳✹✳✷❝✮ t❤➻
❝❤ó♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ x3 + ax2 + bx + c = 0✳
❈❤ù♥❣ ♠✐♥❤✳ ❱➻ x1, x2, x3 ❧➔ ❜❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ♥➯♥ ♣❤➙♥ t➼❝❤ ✤❛ t❤ù❝
t❤➔♥❤ tứ số t ữủ ỗ t tự s ú ✈ỵ✐ ♠å✐

x✿

x3 + ax2 + bx + c = (x − x1 )(x − x2 )(x − x3 )
= x3 − (x1 + x2 + x3 )x2 + (x1 x2 + x2 x3 + x3 x1 )x − (x1 x2 x3 ).
s số ừ ỗ t tự t❛ ✤✐ ✤➳♥ ❝→❝ t➼♥❤ ❝❤➜t ✭✶✳✹✳✷❛✮✱ ✭✶✳✹✳✷❜✮✱ ✭✶✳✹✳✷❝✮✳
❚ø ❜❛ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✭✶✳✹✳✷❛✮✱ ✭✶✳✹✳✷❜✮✱ ✭✶✳✹✳✷❝✮✱ ❦➳t ❤đ♣ ✈ỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ✤è✐ ①ù♥❣
❝õ❛ ♥❣❤✐➺♠✱ t❛ s✉② r❛ ❝→❝ t q ữợ t q rt õ ➼❝❤ ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉
♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✱ ✤÷í♥❣ trá♥✳

▼➺♥❤ ✤➲ ợ ữ tr t õ
1
1
b
1
+
+
=
;
x1 x 2 x3

c
b) T5 = x21 + x22 + x23 = a2 − 2b;
a) T4 =

✭✶✳✹✳✸❛✮
✭✶✳✹✳✸❜✮

c) T6 = (x1 + x2 )2 + (x2 + x3 )2 + (x3 + x1 )2 = 2(a2 − b);
2

2

2

✭✶✳✹✳✸❝✮

2

d) T7 = (x1 − x2 ) + (x2 − x3 ) + (x3 + x1 ) = 2(a − 3b);

✭✶✳✹✳✸❞✮

2

e) T8 = (x1 + x2 )(x2 + x3 ) + (x2 + x3 )(x3 + x1 ) + (x3 + x1 )(x1 + x2 ) = a + b.

1
1
x1 x2 + x2 x3 + x 3 x1
T2

−b
1
+
+
=
=
=
.
x 1 x2 x3
x1 x2 x3
T3
c
T5 = x21 + x22 + x23 = (x1 + x2 + x3 )2 − 2(x1 x2 + x2 x3 + x3 x1 ) = T12 + 2T2 = a2 − 2b.

❈❤ù♥❣ ♠✐♥❤✳
❜✮

✭✶✳✹✳✸❡✮

❛✮

T4 =

❝✮ ❚❛ ❝â

T6 = (x1 + x2 )2 + (x2 + x3 )2 + (x3 + x1 )2 = 2(x21 + x22 + x23 ) + 2(x1 x2 + x2 x3 + x3 x1 )
= 2(T5 + T2 ) = 2(a2 − 2b + b) = 2(a2 − b).
❞✮ ❚❛ ❝â

T7 = (x1 − x2 )2 + (x2 − x3 )2 + (x3 + x1 )2 = 2(x21 + x22 + x23 ) − 2(x1 x2 + x2 x3 + x3 x1 )

= 2(T5 − T2 ) = 2(a2 − 2b − b) = 2(a2 − 3b).
❡✮ ❈✉è✐ ❝ò♥❣

T8 = (x1 + x2 )(x2 + x3 ) + (x2 + x3 )(x3 + x1 ) + (x3 + x1 )(x1 + x2 )
= x21 + x22 + x23 + 3(x1 x2 + x2 x3 + x3 x1 ) = T5 + 3T2 = (a2 − 2b) + 3b = a2 + b.

✼


▼➺♥❤ ✤➲ ✶✳✸✳ ❚❛ ❝â
a) T9 = x31 + x32 + x33 = −a3 + 3ab − 3c;

✭✶✳✹✳✹❛✮

b) T10 = (x1 + x2 − x3 )(x2 + x3 − x1 )(x3 + x1 − x2 ) = a3 − 4ab + 8c;

✭✶✳✹✳✹❜✮

c) T11 = (x1 + x2 )(x2 + x3 )(x3 + x1 ) = −ab + c;

✭✶✳✹✳✹❝✮

d)

❱ỵ✐ ♠å✐ sè t❤ü❝ k, l, t❛ ❝â

T12 = (k + lx1 )(k + lx2 )(k + lx3 ) = k 3 − k 2 la − kl2 b − l3 c.

❈❤ù♥❣ ♠✐♥❤✳

✭✶✳✹✳✹❞✮

❛✮ ❚❛ ❝â

(x1 + x2 + x3 )3 = (x1 + x2 )3 + 3(x1 + x2 )2 x3 + 3(x1 + x2 )x23 + x33
= (x1 + x2 )3 + 2(x1 + x2 )x3 (x1 + x2 + x3 ) + x33
= x31 + 3x1 x2 (x1 + x2 ) + x32 + 3T1 (x1 x3 + x2 x3 ) + x33
= x31 + x32 + x33 + 3T1 (x1 x3 + x2 x3 ) + 3x1 x2 (T1 − x3 )
= x31 + x32 + x33 + 3T1 (x1 x2 + x2 x3 + x3 x1 ) − 3x1 x2 x3
= x31 + x32 + x33 + 3T1 T2 − 3T3 .
❑❤✐ ✤â

T9 = x31 + x32 + x33 = (x1 + x2 + x3 )3 − 3T1 T2 + 3T3
= T13 − 3T1 T2 + 3T3 = −a3 + 3ab − 3c.
❜✮ ❚❛ ❝â

T10 = (x1 + x2 − x3 )(x2 + x3 − x1 )(x3 + x1 − x2 )
= (T1 − 2x1 )(T1 − 2x2 )(T1 − 2x3 )
= T13 − 2T12 (x1 + x2 + x3 ) + 4T1 (x1 x2 + x2 x3 + x3 x1 ) − 8x1 x2 x3
= −T13 + 4T1 T2 − 8T3 = a3 − 4ab + 8c.
❝✮ ❚❛ ❝â

T11 = (x1 + x2 )(x2 + x3 )(x3 + x1 ) = (T1 − x1 )(T1 − x2 )(T1 − x3 )
= T13 − T12 (x1 + x2 + x3 ) + T1 (x1 x2 + x2 x3 + x3 x1 ) − x1 x2 x3
= T1 T2 − T3 = −ab + c.
❞✮ ❚❛ ❝â

T12 = (k + lx1 )(k + lx2 )(k + lx3 ) = k 3 + k 2 l(x1 + x2 + x3 ) + kl2 (x1 x2 + x2 x3 + x3 x1 ) + l3 x1 x2 x3
= k 3 − k 2 la + kl2 b − l3 c.

✽


❍➺ q✉↔ ✶✳✹✳ ❚ø ✭✶✳✹✳✹❞✮ t❛ s✉② r❛ ❤➺ q✉↔ q✉❛♥ trå♥❣ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ t➻♠ r❛ ❝→❝
tự ợ tr t ữ s

(1 x1 )(1 − x2 )(1 − x3 ) = 1 + a + b + c;
(1 + x1 )(1 + x2 )(1 + x3 ) = 1 − a + b − c.

▼➺♥❤ ✤➲ ✶✳✺✳ ❚❛ ❝â
❈❤ù♥❣ ♠✐♥❤✳

a) T13 = x21 x22 + x22 x23 + x23 x21 = b2 − 2ac;

✭✶✳✹✳✺❛✮

b) T14 = x41 + x42 + x43 = a4 − 4a2 b + 2b2 + ac.

✭✶✳✹✳✺❜✮

❚❛ ❝â

❛✮

T13 = x21 x22 + x22 x23 + x23 x21
= (x1 x2 + x2 x3 + x3 x1 )2 − 2(x1 x22 x3 + x1 x2 x23 + x21 x2 x3 )
= (x1 x2 + x2 x3 + x3 x1 )2 − 2x1 x2 x3 (x1 + x2 + x3 )
= T22 − 2T1 T3 = b2 − 2ac.
❜✮

T14 = x41 + x42 + x43
= (x21 + x22 + x23 )2 − 2(x21 x22 + x22 x23 + x23 x21 )
= T52 − 2T13 = (a2 − 2b)2 − 2(b2 − 2ac) = a4 − 4a2 b + 2b2 + 4ac.

▼➺♥❤ ✤➲ ✶✳✻✳ ❚❛ ❝â
x1 + x2 x2 + x3 x3 + x1
ab
+
+
=
− 3.
x3
x1
x2
c
1
1
1
a2 + b
b)T16 =
+
+
=
.
x 1 + x2 x2 + x3 x3 + x1
c − ab
x1
x2
x3
a3 − 2ab + 3c

+
+
=
.
c)T17 =
x2 + x3 x3 + x 1 x1 + x2
ab − c
x1 x2 x2 x3 x3 x1
b2
d)T18 =
+
+
= 2a − .
x3
x1
x2
c
1
1
1
a
+
+
= .
e)T19 =
x1 x2 x2 x3 x3 x1
c
x1
x2
x3

2b − a2
f )T20 =
+
+
=
.
x2 x3 x3 x1 x1 x2
c
1
1
1
b2 2a
g)T21 = 2 + 2 + 2 = 2 − .
x1 x2 x3
c
c
a)T15 =

✾

✭✶✳✹✳✻❛✮

✭✶✳✹✳✻❜✮

✭✶✳✹✳✻❝✮

✭✶✳✹✳✻❞✮

✭✶✳✹✳✻❡✮

✭✶✳✹✳✻❢ ✮

✭✶✳✹✳✻❣✮


❈❤ù♥❣ ♠✐♥❤✳

❚❛ ❝â

❛✮

x1 + x2 x 2 + x3 x3 + x1
+
+
x3
x1
x2
x1 + x2 + x3 x1 + x2 + x3 x1 + x2 + x3
=
+
+
x1
x2
x3
1
1
1
+
+
−3

= (x1 + x2 + x3 )
x1 x2 x3
ab
= T1 T4 − 3 =
− 3;
c

T15 =

❜✮

1
1
1
+
+
x1 + x2 x2 + x3 x3 + x1
x21 + x22 + x23 + 3(x1 x2 + x2 x3 + x3 x1 )
=
(x1 + x2 )(x2 + x3 )(x3 + x1 )
T5 + 3T2
a2 − 2b + 3b
a2 + b
=
=
=
;
T11
−ab + c
c − ab

T16 =

❝✮

x1
x2
x3
+1+
+1+
+1
x2 + x 3
x3 + x1
x1 + x2
x1 + x 2 + x3 x1 + x2 + x3 x 1 + x2 + x3
+
+
=
x2 + x3
x 3 + x1
x1 + x2
1
1
1
= (x1 + x2 + x3 )
+
+
x2 + x3 x3 + x1 x1 + x2
2
a +b

= T1 T16 = (−a)
.
c − ab

T17 + 3 =

❙✉② r❛

T17 =

−a3 − ab
a3 − 2ab + 3c
−3=
.
−ab + c
ab − c

❞✮

x1 x2 x2 x 3 x3 x1
x2 x2 + x22 x23 + x23 x21
+
+
= 1 2
x3
x1
x2
x1 x2 x 3
2
2

T13
b − 2ac
b
=
=
= 2a − .
T3
−c
c

T18 =

❡✮

T19 =

1
1
1
x1 + x2 + x3
T1
a
+
+
=
=
= .
x1 x2 x 2 x3 x3 x1
x1 x2 x3
T3

c

❢✮

T20

x1
x2
x3
x21 + x22 + x23
T5
2b − a2
=
+
+
=
=
=
.
x2 x3 x3 x1 x1 x2
x1 x2 x3
T3
c
✶✵


❣✮

T21 =

T13
1
1
x21 x22 + x22 x23 + x23 x21
b2 2a
1
=
+
+
=
=
− .
x21 x22 x23
x21 x22 x23
T32
c2
c

▼➺♥❤ ✤➲ ✶✳✼✳
T22 = (x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2 = −4a3 c + a2 b2 + 18abc − 4b3 − 27c3 .

❈❤ù♥❣ ♠✐♥❤✳

✭✶✳✹✳✼✮

✣➦t

T (x1 , x2 , x3 ) = (x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2
= (x21 − 2x1 x2 + x22 )(x22 − 2x2 x3 + x23 )(x23 − 2x3 x1 + x21 )
T (x1 , x2 , x3 )

❱➻

❧➔ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ t❤✉➛♥ ♥❤➜t ❜➟❝ ✻ ♥➯♥ t❛ ❝â t❤➸ ♣❤➙♥ t➼❝❤ t❤❡♦ ❝→❝ ✤❛

t❤ù❝ ✤è✐ ①ù♥❣ ❝ì sð

T1 = x1 + x2 + x3 ; T2 = x1 x2 + x2 x3 + x3 x1 ; T3 = x1 x2 x3

♥❤÷ s❛✉

T (x1 , x2 , x3 ) = a1 T16 + a2 T14 T2 + a3 T12 T22 + a4 T23 + a5 T13 T3 + a6 T32 + a7 T1 T2 T3 .
T (x1 , x2 , x3 ) = (x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2
4 ♥➯♥ a1 = a2 = 0. ❑❤✐ ✤â t❛ ❝â

❉♦
❧➔

❝â ❜➟❝ ❝❛♦ ♥❤➜t ✤è✐ ✈ỵ✐ tø♥❣ ❜✐➳♥

x1 , x2 , x3

T (x1 , x2 , x3 ) = a3 T12 T22 + a4 T23 + a5 T13 T3 + a6 T32 + a7 T1 T2 T3 .
a3 , a4 , a5 , a6 , a7 t❛ ❝❤♦ (x1 , x2 , x3 ) ❧➛♥ ❧÷đt ♥❤➟♥ ❝→❝ ❣✐→ trà (0, 1, −1), (0, 1, 1),
(1, −2, 1), (−1, 1, 1), (1, 1, 1). ❱ỵ✐ (x1 , x2 , x3 ) = ((0, −1, 1) t❤➻ T (0, 1, −1) = (0 − 1)2 (1 +
1)2 (−1 − 0)2 = 4.
❚❛ ❧↕✐ ❝â T1 = 0, T2 = −1, T3 = 0. ❉♦ ✤â
✣➸ t➻♠ ❝→❝ ❤➺ sè

T (0, 1, −1) = a3 .0 + a4 (−1)3 + a5 .0 + a6 .0 + a7 .0 = −a4 .

a4 = −4.
−27, a7 = 18. ❑❤✐
❙✉② r❛

❚÷ì♥❣ tỹ ợ trữớ ủ ỏ t ữủ

a3 = 1, a5 = −4, x6 =

✤â

T (x1 , x2 , x3 ) = (x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2
= a3 T12 T22 + a4 T23 + a5 T13 T3 + a6 T32 + a7 T1 T2 T3
= (−a)2 b2 − 4b3 − 4(−a)3 (−c) + 18(−a)b(−c) − 27(−c)2
= a2 b2 − 4b3 − 4a3 c + 18abc − 27c2 .
❱➟②

T22 = (x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2 = −4a3 c + a2 b2 + 18abc − 4b3 − 27c3 .

▼➺♥❤ ✤➲ ✶✳✽✳ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ax
♣❤➙♥ ❜✐➺t x1, x2 t❤➻

3

+ bx2 + cx + d = 0(a = 0)

2

4ac − b
x1 x2 ≥
.

4a2

✶✶

❝â ❤❛✐ ♥❣❤✐➺♠ t❤ü❝


❈❤ù♥❣ ♠✐♥❤✳

x1 , x2

●✐↔ sû

❧➔ ❤❛✐ ♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ✤➣ ❝❤♦✳ ❑❤✐ ✤â t❛

❝â

ax31 + bx21 + cx1 + d = 0;
ax32 + bx22 + cx2 + d = 0.
❚rø ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✱ t❛ ✤÷đ❝

a(x31 − x32 ) + b(x21 − x22 ) + c(x1 − x2 ) = 0.
❱➻

x1 , x2

❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ♥➯♥ ❝❤✐❛ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝❤♦

x1 − x2 , t❛ ✤÷đ❝

a(x21 + x1 x2 + x22 ) + b(x1 + x2 ) + c = 0.
❍❛②

a(x1 + x2 )2 + b(x1 + x2 ) + c − ax1 x2 = 0.
❚❤❡♦ ❣✐↔ t❤✐➳t
2

x1 , x2

tỗ t t õ

= b2 4a(c − ax1 x2 ) ≥ 0,

tù❝ ❧➔

x1 x2 ≥

4ac − b
.
4a2

❍➺ q✉↔ ✶✳✾✳ ❛✮ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ax

3

♣❤➙♥ ❜✐➺t t❤➻

+ b2 + cx + d = 0(a = 0)

❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝

4a2 b − 12ac + 3b2 ≥ 0.

❜✮ ◆➳✉ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ x3 + ax2 + bx + c = 0 ❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝ ♣❤➙♥ ❜✐➺t t❤➻ x1x2 ≥
4b − a2
✈➔ 3a2 + 4a − 12b ≥ 0.
4
❈❤ù♥❣ ♠✐♥❤✳

❛✮ ❱➻ ❜❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ ❧➔ t❤ü❝ ♣❤➙♥ ❜✐➺t ♥➯♥

x1 x2 ≥

4ac − b2
4ac − b2
4ac − b2
;
x
x
≥
;
x
x
≥
.
2
3
3
1
4a2

4a2
4a2

❈ë♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝

x1 x2 + x2 x3 + x3 x1 ≥ 3

4ac − b2
.
4a2

❚❤❡♦ ✭✶✳✹✳✷❜✮ t❛ ❧↕✐ ❝â

T2 = x1 x2 + x2 x3 + x3 x1 = b.
❙✉② r❛

4ac − b2
⇔ 4a2 b − 12ac + 3b2 ≥ 0.
4a2
3
2
♥❣❤✐➺♠ t❤ü❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ x + ax + bx + c = 0.

b≥3
❜✮ ●✐↔ sû

x1 , x2 , x3

❧➔ ❤❛✐

✤â

x31 + ax21 + bx1 + c = 0;

x32 + ax22 + bx2 + c = 0.
✶✷

❑❤✐


❚rø t❤❡♦ ✈➳ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ ✤÷đ❝

(x1 − x2 )3 + a(x1 − x2 )2 + b(x1 − x2 ) = 0.
❱➻

x1 , x2

❧➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ♥➯♥ ❝❤✐❛ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝❤♦

x1 − x2 , t❛ ✤÷đ❝

(x21 + x1 x2 + x22 ) + a(x1 + x2 ) + b = 0
⇔(x1 + x2 )2 + a(x1 + x2 ) + b − x1 x2 = 0.
tt

x1 , x2

tỗ t t❛ ❝â

∆ = a2 − 4(b − x1 x2 ) ≥ 0,

x1 x2 ≥

tù❝ ❧➔

4b − a2
.
4

❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â

x2 x3 ≥

4b − a2
4b − a2
; x3 x1 ≥
.
4
4

❈ë♥❣ ✸ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝

x1 x2 + x2 x3 + x3 x1 ≥ 3.

4b − a2
.
4

❚❤❡♦ ✭✶✳✹✳✷❜✮ t❛ s✉② r❛

b ≥ 3.

4b − a2
⇔ 3a2 + 4a 12b 0.
4

ỵ tr ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✮✳ P❤÷ì♥❣ tr➻♥❤ x
bx + c = 0

✭✈ỵ✐ ❝→❝ ❤➺ sè t❤ü❝ a, b, c✮ ❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝ x1, x2, x3 ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
−4a3 c + a2 b2 + 18abc − 4b3 − 27c2 ≥ 0.

❈❤ù♥❣ ♠✐♥❤✳

x3 + ax2 + bx + c = 0

3

+ ax2 +

✭✶✳✹✳✽✮

a, b, c✮ ❝â ❜❛
2
♥❣❤✐➺♠ t❤ü❝ x1 , x2 , x3 t❤➻ t❤❡♦ ✭✶✳✹✳✼✮ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ (x1 − x2 ) (x2 − x3 ) (x3 − x1 ) =
−4a3 c + a2 b2 + 18abc − 4b3 − 27c3 ≥ 0 ✈ỵ✐ ♠å✐ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t✳
3
2 2
3
2

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû −4a c + a b + 18abc − 4b − 27c ≥ 0 ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤÷ì♥❣ tr➻♥❤
x3 + ax2 + bx + c = 0 ✭✈ỵ✐ ❝→❝ ❤➺ sè t❤ü❝ a, b, c✮ ❝â ❜❛ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t✱ ♠ët ♥❣❤✐➺♠ t❤ü❝
x1 ✈➔ ❤❛✐ ♥❣❤✐➺♠ ♣❤ù❝ x2 = A + Bi, x3 = A − Bi, B = 0 õ t õ
ữỡ tr

ợ sè t❤ü❝

2

2

(x1 − x2 )(x2 − x3 )(x3 − x1 ) = 2Bi (x1 − A)2 + B 2 .
❙✉② r❛

(x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2 = −4B 2 (x1 − A)2 + B 2
♠➙✉ t❤✉➝♥ ❣✐↔ t❤✐➳t

2

< 0,

(x1 − x2 )2 (x2 − x3 )2 (x3 − x1 )2 = −4a3 c + a2 b2 + 18abc − 4b3 − 27c3 ≥ 0.

❉♦ ✤â ♥➳✉✭✶✳✹✳✽✮ ✤ó♥❣ t❤➻ ❝↔ ❜❛ ♥❣❤✐➺♠ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠ t❤ü❝✳

✶✸


ỵ Pữỡ tr x

3

t tự

+ ax2 + bx + c = 0

❝â ❜❛ ♥❣❤✐➺♠ ❞÷ì♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t❛ ❝â

a < 0, b > 0, c < 0.

❈❤ù♥❣ ♠✐♥❤✳

◆➳✉ ♣❤÷ì♥❣ tr➻♥❤

✭✶✳✹✳✾✮

x3 + ax2 + bx + c = 0 ❝â ❜❛ ♥❣❤✐➺♠ ❞÷ì♥❣ t❤➻ ❝❤ó♥❣ ♣❤↔✐ ❧➔

♥❤ú♥❣ ♥❣❤✐➺♠ t❤ü❝ ♥➯♥ t❤❡♦ ✣à♥❤ ❧➼ ✶✳✶✵ t❛ ❝â ✤➥♥❣ t❤ù❝ ✭✶✳✹✳✽✮ ✈➔ t❤❡♦ ✭✶✳✹✳✷❛✮✱ ✭✶✳✹✳✷❜✮
✈➔ ✭✶✳✹✳✷❝✮ t❛ ❝â

x1 + x2 + x3 = −a > 0; x1 x2 + x2 x3 + x3 x1 = b > 0; x1 x2 x3 = −c > 0.
❑❤✐ ✤â✱ t❛ ✤÷đ❝ ✭✶✳✹✳✾✮✳

x3 + ax2 +
bx+c = 0 ❝â ❜❛ ♥❣❤✐➺♠ t❤ü❝✳ ●✐↔ sû x1 ≤ 0 ❦➳t ❤đ♣ ✈ỵ✐ ✭✶✳✹✳✾✮✱ t❛ s✉② r❛ x31 +ax21 +bx1 +c < 0.
3
2
❈❤ù♥❣ tä x1 ≤ 0 ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❝õ❛ x + ax + bx + c = 0✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t✳
3

2
❱➟② ♥➳✉ ❝â ✭✶✳✹✳✽✮ ✈➔ ✭✶✳✹✳✾✮ t❤➻ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ x + ax + bx + c = 0 ♣❤↔✐ ❧➔ ♥❤ú♥❣ sè
◆❣÷đ❝ ❧↕✐✱ ♥➳✉ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✳✽✮ t❤➻ t❤❡♦ ✣à♥❤ ❧➼ ✶✳✶✵✱ ♣❤÷ì♥❣ tr➻♥❤

❞÷ì♥❣✳

✣à♥❤ ỵ ừ ữỡ tr x

3

+ ax2 + bx + c = 0

t❛♠ ❣✐→❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ t❛ ❝â ✭✶✳✹✳✽✮ ✈➔ ✭✶✳✹✳✾✮ ✈➔

❧➔ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛

a3 − 4ab + 8c > 0.

❈❤ù♥❣ ♠✐♥❤✳

✭✶✳✹✳✶✵✮

❚❤❡♦ ✭✶✳✸✳✶✶✮ t❛ ❝â

(x1 + x2 − x3 )(x2 + x3 − x1 )(x3 + x1 − x2 ) = a3 − 4ab + 8c.
◆➳✉ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

x3 + ax2 + bx + c = 0

❧➔ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝ t❤➻

❝❤ó♥❣ ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣ ✈➔ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝

(x1 + x2 − x3 ) > 0, (x2 + x3 − x1 ) > 0, (x3 + x1 − x2 ) > 0.
❙✉② r❛

(x1 + x2 − x3 )(x2 + x3 − x1 )(x3 + x1 − x2 ) = a3 − 4ab + 8c > 0.

✣✐➲✉ ♥➔② ❝❤ù♥❣ tä

✭✶✳✹✳✽✮✱ ✭✶✳✹✳✾✮ ✈➔ ✭✶✳✹✳✶✵✮ t❤ä❛ ♠➣♥✳
◆❣÷đ❝ ❧↕✐✱ t❛ ❝â ✭✶✳✹✳✽✮ ✈➔ ✭✶✳✹✳✾✮ s✉② r❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛

0

x3 +ax2 +bx+c =

❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣✳

❚❤❡♦ ✭✶✳✹✳✶✵✮✱ t❛ ❝â

(x1 + x2 − x3 )(x2 + x3 − x1 )(x3 + x1 − x2 ) = a3 − 4ab + 8c > 0.
❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤

(x1 + x2 − x3 ) > 0, (x2 + x3 − x1 ) > 0, (x3 + x1 − x2 ) > 0.
✶✹


x1 + x2 − x3 ≤ 0 t❤➻
t❤❡♦ ✭✶✳✹✳✶✵✮ ♣❤↔✐ ❝â ♠ët ❜➜t ✤➥♥❣ t❤ù❝ ♥ú❛ ❝â ❞➜✉ ♥❣÷đ❝ ❧↕✐✱ ✈➼ ❞ö✱ x1 + x3 − x2 ≤ 0. ❈ë♥❣

t❤❡♦ ✈➳ ✷ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t❛ s✉② r❛ x1 ≤ 0✱ ♠➙✉ t❤✉➝♥ ❣✐↔ t❤✐➳t✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä
3
2
❜❛ ♥❣❤✐➺♠ t❤ü❝ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ x + ax + bx + c = 0 t❤ä❛ ♠➣♥ ❝→❝ ❜➜t
❚❤➟t ✈➟②✱ ♥➳✉ ♠ët tr♦♥❣ ❜❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝â ❞➜✉ ♥❣÷đ❝ ❧↕✐✱ ✈➼ ❞ư

✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝✳ ❙✉② r❛ ❝❤ó♥❣ ❧➔ ✤ë ❞➔✐ ❜❛ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝✳

✶✳✺ ▼➮■ ▲■➊◆ ❍➏ ●■Ú❆ ❈⑩❈ ◆●❍■➏▼ ❈Õ❆ ▼❐❚
❙➮ P❍×❒◆● ❚❘➐◆❍ ❇❾❈ ❇❆
❚r♦♥❣ ♠ö❝ ♥➔② t❛ ①➙② ❞ü♥❣ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ♥❣❤✐➺♠ ❝õ❛ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛✳ ❈ư
t❤➸✱ ♥➳✉ ❜✐➳t ❜❛ ♥❣❤✐➺♠ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛ t❛ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët sè ♣❤÷ì♥❣
tr➻♥❤ ❜➟❝ ❜❛ ♥❤➟♥ ❝→❝ ♥❣❤à❝❤ ✤↔♦✱ tê♥❣ ✤ỉ✐ ♠ët✱ ❝→❝ ❜➻♥❤ ♣❤÷ì♥❣✱ ✳ ✳ ✳ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ♥➔②
❧➔♠ ♥❣❤✐➺♠✳

▼➺♥❤ ✤➲ ✶✳✶✸✳ ◆➳✉ x , x , x ❧➔ ❜❛ ♥❣❤✐➺♠ ❝õ❛ ữỡ tr ợ c = 0 t x1 , x1 , x1
1

2

3

1

❧➔ ❜❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

b
a
1
t3 + t2 + t + = 0.

c
c
c

❈❤ù♥❣ ♠✐♥❤✳

c = 0 ♥➯♥ ✭✶✳✹✳✶✮ ❦❤æ♥❣ ❝â
1
1
1
+ a 2 + b + c = 0, tữỡ
3
t
t
t



t ữủ



2

3



x = 0.

x=

1
t

ữỡ tr

ữỡ ợ

ử ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛

x3 + x2 − 3x − 2 = 0.
❚❛ ❝â t❤➸ ❦✐➸♠ tr❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❜❛

−1 −
2

√

5 −1 +
,
2

√

5

✈➔

1
t
❚❤❡♦ ❣✐↔ t❤✐➳t✱

❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥

−1
.
2
3

+

❚❤❛②

1
t

x=

2

−

1
t

2x3 + 3x2 − x − 1 = 0

✈➔♦ ♣❤÷ì♥❣ tr➻♥❤

❝â ❜❛ ♥❣❤✐➺♠ ❧➔

x3 + x2 − 3x − 2 = 0,

t❛ ✤÷đ❝

3
− 2 = 0 ⇔ 2t3 + 3t2 − t − 1 = 0.
t

√
−1 − 5
1
2
√ ;
t1 =
⇔ x1 =
=
2
t1
−1 − 5
√
−1 + 5
1
2
√ ;
t2 =
⇔ x2 =

=
2
t2
−1 + 5
−1
1
t3 =
⇔ x3 =
= −2.
2
t3
2
2
√ ;
√
✤➛✉ ❝â ❜❛ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t ❧➔
−1 − 5 −1 + 5
✶✺

✈➔

−2.