Phương pháp tam thức bậc hai trong giải phương trình và hệ phương trình

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖

✣➱ ❚❍➚ ❑■➋❯ ❚❘❆◆●

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

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè

❍⑨ ◆❐■✱ ✷✵✶✾


❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✖

✣➱ ❚❍➚ ❑■➋❯ ❚❘❆◆●
P❍×❒◆● P❍⑩P ❚❆▼ ❚❍Ù❈ ❇❾❈ ❍❆■
❚❘❖◆● ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❱⑨ ❍➏
P❍×❒◆● ❚❘➐◆❍

❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈
❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ số
ữớ ữợ ồ

ì

▲❮■ ❈❆▼ ✣❖❆◆
❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❜↔♥ t❤➙♥
❡♠✱ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ữợ sỹ ữợ ừ ữỡ r
❝ù✉✱ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè
t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❊♠ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐✿ ✧P❤÷ì♥❣ ♣❤→♣ t❛♠ t❤ù❝

❜➟❝ ❤❛✐ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✧ ❧➔ ❦➳t q✉↔
❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥é ❧ü❝ ❤å❝ t➟♣ ❝õ❛ ❜↔♥ t❤➙♥✱ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐
❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉ s❛✐ ❡♠ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥

✣é ❚❤à ❑✐➲✉ ❚r❛♥❣

✐


▲❮■ ❈❷▼ ❒◆
❑❤â❛ ❧✉➟♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❦❤♦❛ trữớ ồ ữ
ở ữợ sỹ ữợ ồ ừ ữỡ
❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝ỉ ❚❤❙✳ ữỡ
ữớ ữợ st s❛♦ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱
♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❙ü ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ♥❣❤✐➯♠ tó❝
tr♦♥❣ ♥❣❤✐➯♥ ự ỳ ữợ ú ừ ổ t✐➲♥ ✤➲ q✉❛♥
trå♥❣ ❣✐ó♣ ❡♠ ❝â ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥✱ ❝→❝ t❤➛②✱ ❝æ

❣✐→♦ tr♦♥❣ tê ✣↕✐ sè ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷
♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱
♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳
❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ♥➔② ♠➦❝ ❞ò ✤➣ ❝â r➜t ♥❤✐➲✉ ❝è
❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ❜↔♥ t❤➙♥ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛
❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ỳ t sõt rt ữủ sỹ õ
õ ỵ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦✱ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✈➔ ❜↕♥ ✤å❝✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✾
❙✐♥❤ ✈✐➯♥

✣é ❚❤à ❑✐➲✉ ❚r❛♥❣

✐✐


▼Ư❈ ▲Ư❈

▲í✐ ♠ð ✤➛✉
✶✳

❚r❛♥❣
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❑✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐

✶

✳ ✳ ✳ ✳

✸

✶✳✶✳ ◆❣❤✐➺♠ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỵ t tờ qt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳✷✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỵ t ừ t tự ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✹✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✷✳ ❈→❝❤ ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✹✳✸✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỵ ừ t tự ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✶✳✺✳✷✳ ❙♦ s→♥❤ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♠ët t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✈ỵ✐ ♠ët
sè α ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✺✳✸✳ ❙♦ s→♥❤ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♠ët t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✈ỵ✐ ❤❛✐
sè α, β (α < β) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✻✳ ❉➜✉ t❛♠ t❤ù❝ tr➯♥ ♠ët ♠✐➲♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✶✳✻✳✶✳ ❇➔✐ t♦→♥ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✶✳✻✳✷✳ ❇➔✐ t♦→♥ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✶✳✻✳✸✳ ❇➔✐ t♦→♥ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✐✐✐


✷✳

Ù♥❣ ❞ư♥❣ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺
♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✶✳ P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷✳✶✳✷✳ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ q✉② ✈➲ ❜➟❝ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✷✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✷✳✷✳✷✳ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✐✈


▼❐❚ ❙➮ ❑➑ ❍■➏❯ ❱⑨ ❈❍Ú ❱■➌❚ ❚➁❚
R

❚➟♣ ❤ñ♣ ❝→❝ sè t❤ü❝✳

❈▼❘ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
✤♣❝♠ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤
✈♥

✈æ ♥❣❤✐➺♠

✈

é
ỵ ồ t
tự ❤❛✐ ❣✐ú ♠ët ✈à tr➼ q✉❛♥ trå♥❣ ✈➔ ✤÷đ❝ ①✉②➯♥ sốt tr
ữỡ tr t sỡ ởt ữợ rt q trå♥❣ ❝➛♥ ♣❤↔✐ ❧➔♠ ❦❤✐ ❣✐↔✐
t♦→♥ ❧➔ ❧ü❛ ❝❤å♥ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐✳ ▲í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ❤❛② ❦❤✐ ❧ü❛
❝❤å♥ ✤÷đ❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ t❤➼❝❤ ❤đ♣✳ P❤÷ì♥❣ ♣❤→♣ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ❧➔
♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❤✐➺✉ q✉↔ ✈➔ ❝â ù♥❣ ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣
❣✐↔✐ t♦→♥ sì ❝➜♣✳
❱ỵ✐ ỳ ỵ tr ũ ợ ỏ s t tỏ ự ữợ
sỹ ữợ ú ù ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ ❚❤❙✳ ❉÷ì♥❣ ❚❤à ▲✉②➳♥ ❡♠
✤➣ ♠↕♥❤ ❞↕♥ ❝❤å♥ ✤➲ t➔✐✿ ✧P❤÷ì♥❣ ♣❤→♣ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ tr♦♥❣

❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✧ ✤➸ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt
♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ ❣✐ó♣ ❤å❝ s✐♥❤ ❝â ❝→✐ ♥❤➻♥ t♦➔♥ ❞✐➺♥ ❤ì♥
✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ tứ t

ử ự
ữợ q✉❡♥ ✈ỵ✐ ❝ỉ♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ t❤➜② ✤÷đ❝
t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ❝❤✐➳♠ ✈à tr➼ q✉❛♥ trå♥❣ tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ sì ❝➜♣✳

✸✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ tr♦♥❣
❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳

✶


✹✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

✹✳✶✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉
❚❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✈➔ ù♥❣ ❞ö♥❣ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳

✹✳✷✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
❈→❝ ❜➔✐ t♦→♥ ❝â sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❧➼ ❧✉➟♥✱ ♣❤➙♥ t➼❝❤ tê♥❣ ❤đ♣✳

✻✳ ❈➜✉ tró❝ ❝õ❛ ✤➲ t➔✐
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥✱ t➔✐ ❧✐➺✉ t õ ỗ
ữỡ
ã ữỡ tự ỡ t tự
ã ữỡ ❞ư♥❣ tr♦♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✳

✷


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐

✶✳✶✳ ◆❣❤✐➺♠ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐
P❤÷ì♥❣ tr➻♥❤ ax2 + bx + c = 0 (a = 0, a, b, c ∈ R) ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣
tr➻♥❤ ❜➟❝ ❤❛✐✳ ✣❛ t❤ù❝ f (x) = ax2 + bx + c ✤÷đ❝ ❣å✐ ❧➔ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐✳
◆❣❤✐➺♠ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ f (x) ❝ơ♥❣ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ❜➟❝ ❤❛✐ f (x) = 0✳
✣➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ t❛ ❜✐➳♥ ✤ê✐ ♥❤÷ s❛✉
f (x) = ax2 + bx + c = 0 (1)
b

c
b
⇔x + x=− ⇔ x+
a
a
2a
2

2

b2
c
b
= 2 − ⇔ x+
4a
a
2a

2

b2 − 4ac
=
.
4a2

✣➦t ∆ = b2 − 4ac t õ
ã < 0 t ữỡ tr (1) ổ
ã = 0 t ữỡ tr (1) ❝â ♥❣❤✐➺♠ ❦➨♣ x =
• ◆➳✉ ∆ > 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ x1,2

−b
✳
2a

√
−b ± ∆
=
✳
2a

◆❤➟♥ ①➨t
◆➳✉ ❜ ❧➔ sè ❝❤➤♥✱ b = 2b ✱ ∆ = b 2 − ac t❤➻ t❛ ❝â ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠
t❤✉ ồ
ã < 0 ữỡ tr (1) ổ
ã = 0 ♣❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ ❦➨♣ x =
✸

−b
✳
a


• ∆ > 0 ♣❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ x1,2

√
−b
=

a

ỵ t
ỵ t tờ qt
♣❤÷ì♥❣ tr➻♥❤
an xn + an−1 xn−1 + an−2 xn−2 + ... + a2 x2 + a1 x + a0 = 0.

●å✐ x1 , x2 , x3 , ..., xn ❧➔ ♥ ♥❣❤✐➺♠✱ ❦➸ ❝↔ ♥❣❤✐➺♠ ❜ë✐✳ ❑❤✐ ✤â
an−1
x1 + x2 + x3 + ... + xn = −
an
x1 x2 + x1 x3 + ... + xn−1 xn =

an−2
an

x1 x2 x3 + x1 x2 x4 + ... + xn−2 xn−1 xn = −
x1 x2 x3 ...xn = (−1)n

an−3
✱✳✳✳✱
an

a0
✳
an

✣↔♦ ❧↕✐✿ ❈❤♦ ♥ sè ❜➜t ❦ý α1 , α2 , α3 , ..., αn t❤ä❛ ♠➣♥
S1 = α1 + α2 + α3 + ... + αn
S2 = α1 α2 + α1 α3 + ... + αn−1 αn
Sk =

αi1 αi2 ...αik ✭ 1 ≤ i1 < i2 < ik ✮

Sn = α1 α2 α3 ...αn ✳

❑❤✐ ✤â α1 , α2 , α3 , ..., αn ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
X n + S1 X n−1 + S2 X n−2 + ... + (−1)n Sn .

✶✳✷✳✷✳ ❱➼ ❞ư ♠✐♥❤ ❤å❛
❱➼ ❞ư ✶✳✷✳✶✳ ✭❈✣❙P❍◆✲✾✾✮ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤
x2 − 2kx − (k − 1)(k − 3) = 0.

ợ ồ k ữỡ tr ổ õ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 t❤ä❛
1
♠➣♥ (x1 + x2 )2 + x1 x2 − 2(x1 + x2 ) + 3 = 0.
4
✹


▲í✐ ❣✐↔✐✳
❚❛ ❝â ∆ = k 2 + (k − 1)(k − 3) = 2k 2 − 4k + 4 = 2 (k − 1)2 + 2 > 0, ∀k.
❙✉② r❛ ♣❤÷ì♥❣ tr➻♥❤ ❧✉ỉ♥ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 t❤ä❛ ♠➣♥
x1 + x2 = 2k
.
x1 x2
= −(k − 1)(k − 3)
❑❤✐ ✤â
1
1
(x1 + x2 )2 +x1 x2 2(x1 +x2 )+3 = (2k)2 (k1)(k3)2(2k)+3 = 0
4

4


ỵ t ừ t tự
ỵ
t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ f (x) = ax2 + bx + c (a = 0)
• ◆➳✉ ∆ < 0 t❤➻ af (x) > 0, ∀x ∈ R
b
2a
• ◆➳✉ ∆ > 0 t❤➻ f (x) ❝â ❤❛✐ ♥❣❤✐➺♠ x1 ✈➔ x2 (x1 < x2 )
• ◆➳✉ ∆ = 0 t❤➻ af (x) > 0, ∀x = −

✰ af (x) > 0, ∀x ∈
/ (x1 , x2 )
✰ af (x) < 0, x (x1 , x2 )

ị ồ
ã ∆>0

❍➻♥❤ ✶✳✶✿ ❛ ✈➔ ❜

✺


❍➻♥❤ ✶✳✶❜✳ a < 0

❍➻♥❤ ✶✳✶❛✳ a > 0
f (x) > 0 ⇔

f (x) > 0 ⇔ x1 < x < x2

x > x2
x < x1

f (x) < 0 ⇔

f (x) < 0 ⇔ x1 < x < x2 .
• ∆=0

❍➻♥❤ ✶✳✷✿ ❛ ✈➔ ❜

❍➻♥❤ ✶✳✷❛✳ a > 0 : f (x) > 0, ∀x = x0 .
❍➻♥❤ ✶✳✷❜✳ a < 0 : f (x) < 0, ∀x = x0 .
• ∆<0

❍➻♥❤ ✶✳✸✿ ❛ ✈➔ ❜

✻

x > x2
x < x1 .


❍➻♥❤ ✶✳✸❛✳ a > 0 : f (x) > 0, ∀x ∈ R.
❍➻♥❤ ✶✳✸❜✳ a < 0 : f (x) < 0, ∀x ∈ R.

❝✳ ◆❤➟♥ ①➨t
• f (x) > 0 ∀x ∈ R ⇔

• f (x) < 0 ∀x ∈ R ⇔

a>0

• f (x) ≥ 0 ∀x ∈ R ⇔

∆<0
a<0

• f (x) ≤ 0 ∀x ∈ R ⇔

∆<0

❞✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛
❱➼ ❞ö ✶✳✸✳✶✳ ❳➨t ❞➜✉ ❝→❝ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ s❛✉
❛✳ f (x) = 3x2 + x + 5
❜✳ g(x) = −4x2 + 12x − 9
❝✳ h(x) = 12x2 + 2(a + 3)x + a.

▲í✐ ❣✐↔✐✳
❛✳ f (x) ❧➔ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ❝õ❛ x ❝â a = 3 > 0✱ ∆ = −59 < 0.
❱➟② f (x) > 0, ∀x ∈ R.
❜✳ g(x) ❧➔ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ❝õ❛ x ❝â a = −4 < 0✱ ∆ = 0.
3
❱➟② g(x) < 0, ∀x = .
2

❝✳ h(x) ❧➔ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ❝õ❛ x ❝â a = 12 > 0✱ ∆ = (a − 3)2 .
✰ ❚r÷í♥❣ ❤đ♣ ✶✳ ∆ = 0 ⇔ a = 3.
❱➟② h(x) > 0, ∀x = a.
✰ ❚r÷í♥❣ ❤đ♣ ✷✳ ∆ > 0 ⇔ a = 3.

1
a
❙✉② r❛ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ x1 = − ✈➔ x2 = − .
2
6

❳➨t ❤❛✐ ❦❤↔ ♥➠♥❣ s❛✉

❑❤↔ ♥➠♥❣ ✶✳ x1 < x2 ⇔ a < 3.
✼

a>0
∆≤0
a<0
∆≤0


❑❤✐ ✤â



f (x) > 0 ⇔ 


x>−

a
6

x<−

1
2

1
a
f (x) < 0 ⇔ − < x < −
2
6

❑❤↔ ♥➠♥❣ ✷✳ x1 > x2 ⇔ a > 3.
❑❤✐ ✤â



f (x) > 0 ⇔ 


x>−

1
2

x<−

a
6

a

1
f (x) < 0 ⇔ − < x < −
6
2

❱➼ ❞ö ✶✳✸✳✷✳ ❈❤♦ t❛♠ t❤ù❝ f (x) = (m + 1) x2 − 2(m − 1)x + 3m − 3.
❛✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ m t❤➻ f (x) < 0 ✈ỵ✐ ♠å✐ x.
❜✳ ❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ m t f (x) 0 ợ ồ x.

ớ
ã ợ m + 1 = 0 ⇔ m = −1✱ t❛ ❝â f (x) = 4x − 6✱ ❞♦ ✤â ❦❤æ♥❣ t❤➸ ❝â
f (x) < 0 ✈ỵ✐ ♠å✐ ① ❤♦➦❝ f (x) 0 ợ ồ x.
ã ợ m + 1 = 0 ⇔ m = −1.

❛✳ f (x) < 0 ✈ỵ✐ ♠å✐ x
a<0
m+1<0
⇔
⇔
⇔ m < −2.
∆ <0
2 (m − 1) (m + 2) > 0
❱➟②✱ ✈ỵ✐ m < −2 t❤➻ f (x) < 0 ✈ỵ✐ ♠å✐ x✳
❜✳ f (x) ≥ 0 ✈ỵ✐ ♠å✐ x
a>0
m+1>0
⇔
⇔
⇔ m ≥ 1.
∆ ≤0

2 (m − 1) (m + 2) ≥ 0
❱➟②✱ ✈ỵ✐ m ≥ 1 t❤➻ f (x) ≥ 0 ✈ỵ✐ ♠å✐ x✳
✽


✶✳✹✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐
✶✳✹✳✶✳ ✣à♥❤ ♥❣❤➽❛
❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ ❧➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ f (x) > 0 ❤♦➦❝
f (x) ≥ 0 ❤♦➦❝ f (x) < 0 ❤♦➦❝ f (x) ≤ 0✳ ❚r♦♥❣ ✤â f (x) ❧➔ ♠ët t❛♠ t❤ù❝

❜➟❝ ❤❛✐✳

✶✳✹✳✷✳ ❈→❝❤ ❣✐↔✐
✰ ❳➨t ❞➜✉ f (x) = ax2 + bx + c✳
✰ ▲ü❛ ❝❤å♥ x ✤➸ f (x) > 0 ❤♦➦❝ f (x) ≥ 0 ❤♦➦❝ f (x) < 0 ❤♦➦❝ f (x) ≤ 0✳

✶✳✹✳✸✳ ❱➼ ❞ư ♠✐♥❤ ❤å❛
❱➼ ❞ư ✶✳✹✳✶✳ ❈❤♦ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ m (m + 2) x2 + 2mx + 2 > 0
t ữỡ tr ợ m = 1
ợ ❣✐→ trà ♥➔♦ ❝õ❛ m t❤➻ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ✤ó♥❣ ✈ỵ✐ ♠å✐ x✳

▲í✐ ❣✐↔✐✳
❛✳ ❱ỵ✐ m = 1✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ 3x2 + 2x + 2 > 0✳
∆f = −5 < 0
✣➦t f (x) = 3x2 + 2x + 2 ❝â
⇒ f (x) > 0, ∀x ∈ R
a=3>0
t ữỡ tr õ ợ ồ x.
õ ∆ = m2 − 2m (m + 2) = −m2 − 4m.
✰ ◆➳✉ m = 0 t❤➻ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ✤ó♥❣ ✈ỵ✐ ♠å✐ x✳

✰ ◆➳✉ m = −2 t❤➻ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ −4x+2 > 0✱ ❦❤ỉ♥❣ ♥❣❤✐➺♠
✤ó♥❣ ✈ỵ✐ ♠å✐ x✳
✰ ◆➳✉ m = 0 ✈➔ m = 2 t t
ữỡ tr ú ợ ồ x

m < −2




 m>0
m (m + 2) > 0
m < −4
⇔
⇔
⇔
.

∆ = −m2 − 4m < 0
m
<
−4
m
>
0




 m>0

❱➟② ✈ỵ✐ m < −4 ❤♦➦❝ m ≥ 0 t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳
✾


❱➼ ❞ư ✶✳✹✳✷✳ ❈❤♦ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ x2 + 4x + 3 + m ≤ 0. ❱ỵ✐ ❣✐→ trà ♥➔♦
❝õ❛ m t❤➻
❛✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✈ỉ ♥❣❤✐➺♠✳
❜✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠✳
❝✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❧➔ ♠ët ✤♦↕♥ ❝â ✤ë ❞➔✐ ❜➡♥❣ ✷✳

▲í✐ ❣✐↔✐✳
❛✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✈æ ♥❣❤✐➺♠ ⇔ ∆ < 0 ⇔ 1 − m < 0 ⇔ m > 1.
❜✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠
⇔ ∆ = 0 ⇔ 1 − m = 0 ⇔ m = 1.

❝✳ ✣➸ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ ❧➔ ♠ët ✤♦↕♥ tr➯♥ trö❝ sè ❝â ✤ë ❞➔✐ ❜➡♥❣
✷ t❤➻ t❛♠ t❤ù❝ ð ✈➳ tr→✐ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ♣❤↔✐ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥
❜✐➺tx1 ✈➔ x2 t❤ä❛ |x1 − x2 | = 2

∆ >0
1−m>0
√
⇔
⇔ √
⇔ m = 3.


1

m

=
2

=2
a

ỵ ừ t tự
ỵ
ỵ
t tự f (x) = ax2 + bx + c (a = 0)✳ ◆➳✉ ❝â sè α s❛♦ ❝❤♦
af (α) < 0 t❤➻ t❛♠ t❤ù❝ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 (x1 < x2 ) ✈➔
x1 < α < x2 ✳ ứ ỵ tr t t r

af () < 0 ⇒ f (x) = 0 ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 (x1 < x2 )
✈➔ x1 < α < x2 ✳
❜✳ ◆➳✉ af (α) = 0 ⇒ α ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ t❛♠ t❤ù❝✳
❝✳ ◆➳✉ af (α) > 0✱ ❦❤✐ ✤â ❝➛♥ q✉❛♥ t➙♠ tỵ✐ ❜✐➺t sè ∆ ❝õ❛ t❛♠ t❤ù❝✳
✰ ◆➳✉ ∆ < 0 ⇒ f (x) = 0 ✈æ ♥❣❤✐➺♠ ♥➯♥ ✈✐➺❝ s♦ s→♥❤ sè α ✈ỵ✐ ❝→❝ ♥❣❤✐➺♠
❝õ❛ t❛♠ t❤ù❝ ✈ỉ ♥❣❤➽❛✳
✰ ◆➳✉ ∆ ≥ 0 ⇒ f (x) = 0 ❝â ♥❣❤✐➺♠ ✈➔ sè α ♥➡♠ ♥❣♦➔✐ ❦❤♦↔♥❣ ❤❛✐
♥❣❤✐➺♠✳
✶✵


• ◆➳✉
•

−b
S
=

> α t❤➻ α < x1 ≤ x2
2
2a

S
< α t❤➻ x1 ≤ x2 < α✳
2

❜✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛
❱➼ ❞ư ✶✳✺✳✶✳ ❑❤ỉ♥❣ t➼♥❤ ∆ ❝❤ù♥❣ tä ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✤➙② ❝â ❤❛✐
♥❣❤✐➺♠
❛✳ x2 − 27x + 3 = 0✳
❜✳ m2 + 1 x2 − 2x − m2 − 1 = 0✳

▲í✐ ❣✐↔✐✳
❛✳ ❚❛ ❝â af (1) = 1 − 27 + 3 = −23 < 0✳
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 (x1 < x2 ) ✈➔
x1 < 1 < x2 ✳

❜✳ ❚❛ ❝â af (0) = −m2 − 1 < 0 ✈ỵ✐ ♠å✐ m✳
❱➟② ♣❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ♣❤➙♥ ❜✐➺t x1 , x2 (x1 < x2 ) ✈➔
x1 < 0 < x2 ✳

✶✳✺✳✷✳ ❙♦ s→♥❤ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♠ët t❛♠ t❤ù❝ ❜➟❝ ợ ởt
số
trữớ ủ õ t r❛
❈❤♦ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ f (x) = ax2 + bx + c, (a = 0)✱ sè t❤ü❝ α✱ f (x)
❝â ❤❛✐ ♥❣❤✐➺♠ t❤ä❛ ♠➣♥
✶✮ x1 < α < x2 ⇔ af (α) < 0



∆>0



✷✮ x1 < x2 < α ⇔ af (α) > 0


S

 <α
2

 f (α) = 0
✹✮ x1 < α = x2 ⇔
 S <α
2



∆>0



✸✮ α < x1 < x2 ⇔ af (α) > 0


S

 >α

2

 f (α) = 0
✺✮ x1 = α < x2 ⇔
 S >α
2
✶✶




∆≥0



✻✮ x1 ≤ x2 < α ⇔ af (α) > 0


S

 <α
2



∆≥0



✼✮ α < x1 ≤ x2 ⇔ af (α) > 0

S

>
2

ú ỵ
f (x) = ax2 + bx + c ❝â ♥❣❤✐➺♠ x ∈ D ✭D ❧➔ ♠ët ❦❤♦↔♥❣✱ ♠ët ✤♦↕♥✱ ♥û❛

❦❤♦↔♥❣✱ ♥û❛ ✤♦↕♥✮✳


α < x1 ≤ x2

✶✮f (x) = 0 ❝â ♥❣❤✐➺♠ x > α ⇔ 
 x1 < α < x2
x1 = α < x2


x1 ≤ x2 < α

✷✮f (x) = 0 ❝â ♥❣❤✐➺♠ x < α ⇔ 
 x1 < α < x2
x1 < x2 = α

✸✮f (x) = 0 ❝â ♥❣❤✐➺♠ x ≥ α ⇔

✹✮f (x) = 0 ❝â ♥❣❤✐➺♠ x ≤ α ⇔

α ≤ x1 ≤ x2
x1 ≤ α ≤ x2
x1 ≤ x2 ≤ α
x1 ≤ α ≤ x2

✺✮f (x) = 0 ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠ x > α ⇔

✻✮f (x) = 0 ❝â ✤ó♥❣ ♠ët ♥❣❤✐➺♠ x < α ⇔

x1 ≤ α < x2
α < x1 = x2
x1 < α ≤ x2
x1 = x2 < α

✳

✯ ◆➳✉ f (x) ❝â ❝❤ù❛ t❤❛♠ sè t❤➻ t❛ ①➨t t❤➯♠ tr÷í♥❣ ❤đ♣ a = 0✳

✶✷


❜✳ ❱➼ ❞ư ♠✐♥❤ ❤å❛
❱➼ ❞ư ✶✳✺✳✷✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤
(m + 1) x2 − 2 (m − 1) x + m2 + 4m − 5 = 0.

❱ỵ✐ ❣✐→ trà ♥➔♦ ❝õ❛ ♠ t❤➻
❛✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ tr→✐ ❞➜✉✳
❜✳ Pữỡ tr õ ợ ỡ
Pữỡ tr➻♥❤ ❝â ❤❛✐ ♥❣✐➺♠ ✤➲✉ ♥❤ä ❤ì♥ ✶✳

▲í✐ ❣✐↔✐✳
❛✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ tr→✐ ❞➜✉
⇔ x1 < 0 < x2 ⇔ af (0) < 0
m < −5

⇔ (m + 1) . m2 + 4m − 5 < 0 ⇔

−1 < m < 1

✳

❜✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ✤➲✉ ❧ỵ♥ ❤ì♥ ✷




(m − 1) m2 + 5m + 6 ≤ 0
∆ ≥0






2
af
(2)
>
0
⇔ 2 < x1 ≤ x2 ⇔

⇔ (m + 1) m + 4m + 3 > 0




m+3
S


 >2

<0
m+1
2

⇔ −1 < m ≤ 1.

❝✳ P❤÷ì♥❣ tr➻♥❤ ❝â ❤❛✐ ♥❣❤✐➺♠ ✤➲✉ ♥❤ä ❤ì♥ ✶




(m − 1) m2 + 5m + 6 ≤ 0
∆ ≥0







2
af
(1)
>
0
⇔ x1 ≤ x2 < 1 ⇔
⇔ (m + 1) m + 3m − 2 > 0




−2
S


 <1

<0
m+1
2

√
−3 + 17
.
⇔ −1 < m <
2

✶✸

✶✳✺✳✸✳ ❙♦ s→♥❤ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ♠ët t❛♠ t❤ù❝ ❜➟❝ ợ
số , ( < )
ứ ỗ t❤à ❝õ❛ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐ ✈➔ ✤à♥❤ ❧➼ ✈➲ ❞➜✉ ❝õ❛ t❛♠ t❤ù❝
❜➟❝ ❤❛✐ t❛ ❝â ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉

✶✮ α < x1 < x2 < β ⇔




∆>0




 af (α) > 0

af (β) > 0




S

α< <β
2



∆>0




✷✮ x1 < x2 < α < β ⇔ af (α) > 0


S

 <α
2

✸✮ x1 < α < x2 < β ⇔

af (α) < 0

✹✮ x1 < α < β < x2 ⇔

af (α) < 0

af (β) > 0

af (β) < 0



∆>0



✺✮ α < β < x1 < x2 ⇔ af (β) > 0



S

 >β
2

✻✮ α < x1 < β < x2 ⇔

af (α) > 0

✼✮ x1 = α < β < x2 ⇔

f (α) = 0

af (β) < 0

af (β) < 0
✶✹


✽✮ x1 = α < β = x2 ⇔

f (α) = 0
f (β) = 0



f (α) = 0




✾✮ α = x1 < x2 < β ⇔ af (β) > 0


S

 <β
2


f (β) = 0



✶✵✮ α < β = x1 < x2 ⇔ af (α) > 0


S

 >β
2

 f (α) = 0
✶✶✮ x1 < x2 = α < β ⇔
 S <α
2


f (β) = 0




✶✷✮ α < x1 < β = x2 ⇔ af (α) > 0


S

 <β
2

✶✸✮ x1 < α < β = x2 ⇔

f (β) = 0
af (α) < 0

ú ỵ
f (x) = ax2 + bx + c õ ♥❣❤✐➺♠ x ∈ D ✭D ❧➔ ♠ët ❦❤♦↔♥❣✱ ♠ët ✤♦↕♥✱ ♥û❛

❦❤♦↔♥❣✱ ♥û❛ ✤♦↕♥✮✳


x1 ≤ α ≤ x2 ≤ β

✶✮ f (x) = 0 ❝â ♥❣❤✐➺♠ x ∈ [α, β] ⇔ 
 α ≤ x1 ≤ β ≤ x2
α ≤ x1 ≤ x2 ≤ β

✶✺



α < x1 < β ≤ x2

✷✮ f (x) = 0 ❝â ♥❣❤✐➺♠ x ∈ (α, β) ⇔ 
 x1 ≤ α < x2 < β
α < x1 ≤ x2 < β


x1 ≤ α < x2 < β

✸✮ f (x) = 0 ❝â ✤ó♥❣ ✶ ♥❣❤✐➺♠ x ∈ (α, β) ⇔ 
 α < x1 < β ≤ x2
α < x1 = x2 < β


x = α < β < x2
 1
α≤x =x ≤β

1
2

✹✮ f (x) = 0 ❝â ✤ó♥❣ ✶ ♥❣❤✐➺♠ x ∈ [α, β] ⇔ 
 x1 < α < x2 = β

 α < x1 ≤ β < x2

x1 < α ≤ x2 < β



x ≤ α < x2 < β
 1
α1
2

✺✮ f (x) = 0 ❝â ✤ó♥❣ ✶ ♥❣❤✐➺♠ x ∈ (−∞, α)∪[β, +∞) ⇔ 
 β ≤ x1 = x2

x1 = x2 ≤ α

✻✮ f (x) = 0 ❝â ♥❣❤✐➺♠

x>β
x<α

❤♦➦❝ ✭

x≥β
x≤α

✮✳

❳➨t ❜➔✐ t♦→♥ ♥❣÷đ❝ f (x) = 0 ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ t❤ä❛ ♠➣♥

x>β
x<α

x≥β
x≤α

❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) ✈æ ♥❣❤✐➺♠ ❤♦➦❝ f (x) ❝â ♥❣❤✐➺♠ t❤ä❛ ♠➣♥
α < x1 ≤ x2 < β ✳

✯ ◆➳✉ f (x) ❝â ❝❤ù❛ t❤❛♠ sè t❤➻ ①➨t t❤➯♠ tr÷í♥❣ ❤đ♣ a = 0✳
✶✻

❤♦➦❝


❜✳ ❱➼ ❞ö ♠✐♥❤ ❤å❛
❱➼ ❞ö ✶✳✺✳✸✳ ❳→❝ ✤à♥❤ m ✤➸ ♣❤÷ì♥❣ tr➻♥❤
(m + 1) x2 − 3mx + 4m = 0 (1) ❝â ♥❣❤✐➺♠ t❤✉ë❝ [0, 1] .

▲í✐ ❣✐↔✐✳
✰ ❚r÷í♥❣ ❤đ♣ ✶✳ ◆➳✉ m + 1 = 0 ⇔ m = −1.

4
✭❧♦↕✐✮✳
3

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ (1) trð t❤➔♥❤ −3x + 4 = 0 ⇔ x =
❱➟② m = −1 ❦❤ỉ♥❣ t❤ä❛ ♠➣♥✳
✰ ❚r÷í♥❣ ❤đ♣ ✷✳ ◆➳✉ m + 1 = 0 ⇔ m = −1✳
P❤÷ì♥❣ tr➻♥❤ ✭✶✮ ❝â ♥❣❤✐➺♠ t❤✉ë❝ [0, 1]


x1 ≤ 0 ≤ x2 ≤ 1 (i)


⇔
 0 ≤ x1 ≤ 1 ≤ x2 (ii)
0 ≤ x1 ≤ x2 ≤ β (iii)

• (i) ⇔

af (0) ≤ 0
af (1) ≥ 0

⇔

1
⇔ − ≤ m ≤ 0.
2

• (ii) ⇔

af (0) ≥ 0
af (1) ≤ 0

⇔



−1 ≤ m ≤ 0



(m + 1) 4m ≤ 0

m ≤ −1
⇔


(m + 1) (2m + 1) ≥ 0

1

 m≥−
2

(m + 1) 4m ≥ 0
(m + 1) (2m + 1) 0



ổ ỵ

ã (iii)




0




af (α) ≥ 0


af (β) ≥ 0




S

α≤ ≤β
2

⇔



m2 − 8m − 8 ≥ 0





 (m + 1) 4m ≥ 0

(m + 1) (2m + 1) ≥ 0




3m



≤1
0≤
2 (m + 1)

1
❑➳t ❤đ♣ ✭✐✮✱ ✭✐✐✮✱ ✭✐✐✐✮ t❛ ✤÷đ❝ − ≤ m ≤ 0.
2








m0

m 1


1

1 m
2

ổ ỵ


1
❱➟② ✈ỵ✐ − ≤ m ≤ 0 t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳

2

❱➼ ❞ư ✶✳✺✳✹✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ x2 + 2 (2m − 1) x + m + 1 = 0)✳ (2)
❳→❝ ✤à♥❤ m ✤➸ ♣❤÷ì♥❣ tr➻♥❤ ❝â ♥❣❤✐➺♠ t❤ä❛ ♠➣♥ |x| ≥ 1.

▲í✐ ❣✐↔✐✳
❳➨t ❜➔✐ t♦→♥ ♥❣÷đ❝ ✧❚➻♠ ✤✐➲✉ ❦✐➺♥ m ✤➸ ♣❤÷ì♥❣ tr➻♥❤ (2) ✈ỉ ♥❣❤✐➺♠ ❤♦➦❝
♣❤÷ì♥❣ tr➻♥❤ (2) ❝â ❤❛✐ ♥❣❤✐➺♠ t❤✉ë❝ (−1, 1)✧✳
5
• (2) ✈ỉ ♥❣❤✐➺♠ ⇔ ∆ < 0 ⇔ 4m2 − 5m < 0 ⇔ 0 < m < ✳ (∗)
4



∆ ≥0




 af (−1) > 0
• (2) ❝â ❤❛✐ ♥❣❤✐➺♠ t❤✉ë❝ (−1, 1) ⇔ −1 < x1 ≤ x2 < 1 ⇔

af (1) > 0




S

 −1 < < 1

2

⇔



4m2 − 5m ≥ 0





 4 − 3m > 0

5m > 0





−2 (2m + 1)

 −1 <
<1
2

m0




5



m


4




ổ ỵ ()
4


m<


3

m>0




1 < m < 0

5
t ủ () ✈➔ (∗∗) t❛ ✤÷đ❝ 0 < m < ✳
4
5
❱➟② ✤➸ (2) ❝â ♥❣❤✐➺♠ t❤ä❛ ♠➣♥ |x| ≥ 1 ❦❤✐ m ≤ 0 ❤♦➦❝ m ≥ ✳
4

✶✳✻✳ ❉➜✉ t❛♠ t❤ù❝ tr➯♥ ♠ët ♠✐➲♥
✶✳✻✳✶✳ ❇➔✐ t♦→♥ ✶
❈❤♦ t❛♠ t❤ù❝ f (x) = ax2 + bx + c (a = 0) .
❛✳ ❚➻♠ ✤✐➲✉ ❦✐➺♥ ✤➸ f (x) ≥ 0 ∀x✳
❜✳ ❚➻♠ ✤✐➲✉ ❦✐➺♥ ✤➸ f (x) ≤ 0 ∀x✳
✶✽