❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛ ❡q✉❛çã♦ ❞♦ ✷➸ ❣r❛✉
❍❡r♠❡s ❆♥tô♥✐♦ P❡❞r♦s♦
Pr♦❢❡ss♦r ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ❯◆❊❙P ✕ ❈❛♠♣✉s ❞❡ ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦
❤❡r♠❡s❅✐❜✐❧❝❡✳✉♥❡s♣✳❜r
❘❡s✉♠♦
Pr♦❜❧❡♠❛s q✉❡ r❡❝❛❡♠ ♥✉♠❛ ❡q✉❛çã♦ ❞♦ ✷➸ ❣r❛✉ ❥á ❛♣❛r❡❝✐❛♠✱ ❤á ♠❛✐s ❞❡ q✉❛tr♦ ♠✐❧ ❛♥♦s ❡♠ t❡①t♦s ❡s❝r✐t♦s ❡♠ ♣❧❛❝❛s ❞❡ ❛r❣✐❧❛ ♥❛ ▼❡s♦♣♦tâ♠✐❛✱ ❡ ❡♠ ♣❛♣✐r♦s ♥♦ ❊❣✐t♦✳ ❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ❛❝♦♠♣❛♥❤❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❛ ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♠❛t❡♠át✐❝❛ ❛tr❛✈és ❞♦s ❞✐✈❡rs♦s ♠ét♦❞♦s ❞❡ s♦❧✉çõ❡s✱ ❞❡s❞❡ ❛s r❡❝❡✐t❛s ❡♠ ♣r♦s❛✱ q✉❡ ❡♥s✐♥❛✈❛♠ ❝♦♠♦ ♣r♦❝❡❞❡r ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛s r❛í③❡s✱ ❡♠ ❡①❡♠♣❧♦s ❝♦♥❝r❡t♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥✉♠ér✐❝♦s✱ ❛té ❛ ❢❛♠♦s❛ ❢ór♠✉❧❛ ❣❡r❛❧ ❞❡ r❡s♦❧✉çã♦✳ ❋ór♠✉❧❛ ❡ss❛✱ q✉❡ ❛❞q✉✐r✐✉ ♦ ❛s♣❡❝t♦ q✉❡ t❡♠ ❤♦❥❡✱ s♦♠❡♥t❡ q✉❛♥❞♦ s❡ ❣❡♥❡r❛❧✐③♦✉ ♦ ✉s♦ ❞❡ ❧❡tr❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡ ✉♠❛ ❡q✉❛çã♦✱ ❛ ♣❛rt✐r ❞♦s tr❛❜❛❧❤♦s ❞❡ ❋r❛♥ç♦✐s ❱✐èt❡ ✭✶✺✹✵✲✶✻✵✸✮ ❡ ❞❡ ❘❡♥é ❉❡s❝❛rt❡s ✭✶✺✾✻✲✶✻✺✵✮✳ ❉❡ss❡ ♠♦❞♦✱ ♦ ♣r♦♣ós✐t♦ é r❡❝♦♥st✐t✉✐r ♣♦♥t♦s ✐♠♣♦rt❛♥t❡s ❞♦ ❛ss✉♥t♦ ❞❡s❞❡ ♦s ♠❡s♦♣♦tâ♠✐♦s ❡ ❡❣í♣❝✐♦s ❛té ♦s ❞✐❛s ❛t✉❛✐s✳ P❛r❛ ✐ss♦ ❤á q✉❡ s❡ ❞❡st❛❝❛r ♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❊✉❝❧✐❞❡s ✭✸✵✵ ❛✳❈✳✮ ❡♠ ✏❖s ❊❧❡♠❡♥t♦s✑ q✉❡ ❞❡❞✐❝♦✉ ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s s♦❜r❡ ❝♦♥str✉çõ❡s ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡ ár❡❛s ❡ s♦❜r❡ ♦ s❡❣♠❡♥t♦ á✉r❡♦✱ q✉❡ s❡ ❝♦♠♣♦rt❛♠ ❝♦♠♦ ❝❛s♦s tí♣✐❝♦s ❞❡ ❡q✉❛çõ❡s ❞♦ ✷➸ ❣r❛✉✳ ❆ s❡❣✉✐r t❡♠✲s❡ ❛ ❣r❛♥❞❡ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❤✐♥❞✉s ❡ ár❛❜❡s✱ q✉❡ ✐♥tr♦❞✉③✐r❛♠✱ ♣♦r ♠❡✐♦ ❞❡ r❡❝❡✐t❛s ❡ ❞❡ ❢♦r♠❛ ❣❡♦♠étr✐❝❛✱ ♦ ✐♠♣♦rt❛♥t❡ ♠ét♦❞♦ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s✱ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ s❡ ❝❤❡❣❛r à ❢ór♠✉❧❛ ❝❧áss✐❝❛✳ ❋✐♥❛❧♠❡♥t❡ t❡♠✲s❡ ❛s r❡s♦❧✉çõ❡s ❞❡ ❱✐èt❡ ❡ ❞❡ ❉❡s❝❛rt❡s q✉❡ ♣♦❞❡♠ s❡r ❝❤❛♠❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡ ❛❧❣é❜r✐❝❛ ❡ ❞❡ ❛♥❛❧ít✐❝❛✳ ◆❡ss❡s ❞♦✐s ❝❛s♦s✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❝♦♠♦ s❡ ❢❛③ ❛t✉❛❧♠❡♥t❡✱ ❥á s❡ ✉s♦✉ ❧❡tr❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛s ❡q✉❛çõ❡s✳
P❛❧❛✈r❛s ❝❤❛✈❡s✿ ❍✐stór✐❛ ❞❛ ➪❧❣❡❜r❛✱ ❊q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❊q✉❛çõ❡s q✉❛❞rát✐❝❛s✳
❆ ❜r✐❡❢ ❤✐st♦r② ♦❢ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥
❆❜str❛❝t
▼♦r❡ t❤❛♥ ❢♦✉r t❤♦✉s❛♥❞ ②❡❛rs ❛❣♦✱ ♣r♦❜❧❡♠s ✇❤✐❝❤ ②✐❡❧❞ ❛ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ❛❧r❡❛❞② ❛♣♣❡❛r❡❞ ✐♥ t❡①ts ✇r✐tt❡♥ ✐♥ ❝❧❛② t❛❜❧❡ts ✐♥ ▼❡s♦♣♦t❛♠✐❛ ❛♥❞ ✐♥ ♣❛♣②r✉s ✐♥ ❊❣②♣t✳ ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ ♣r❡s❡♥t ✇♦r❦ ✐s t♦ ❢♦❧❧♦✇ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤✐s ✐♠♣♦rt❛♥t ♠❛t❤❡♠❛t✐❝ t♦♦❧ t❤r♦✉❣❤ s❡✈❡r❛❧ s♦❧✉t✐♦♥ ♠❡t❤♦❞s✱ ❢r♦♠ ♣r♦s❡✲st②❧❡ r❡❝✐♣❡s✱ ✇❤✐❝❤ t❛✉❣❤t ❤♦✇ t♦ ✜♥❞ t❤❡ r♦♦ts ✐♥ ❝♦♥❝r❡t❡ ❡①❛♠♣❧❡s ✉s✐♥❣ ♥✉♠❡r✐❝❛❧ ❝♦❡✣❝✐❡♥ts✱ t♦ t❤❡
P❊❉❘❖❙❖✱ ❍✳❆✳ ✲ ✶ ✲ ❯♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛
t❤❡ ✉s❡ ♦❢ ❧❡tt❡rs t♦ r❡♣r❡s❡♥t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛♥ ❡q✉❛t✐♦♥ ✇❛s s♣r❡❛❞ ❜② t❤❡ ✇♦r❦s ♦❢ ❋r❛♥ç♦✐s ❱✐èt❡ ✭✶✺✹✵✲✶✻✵✸✮ ❛♥❞ ❘❡♥é ❉❡s❝❛rt❡s ✭✶✺✾✻✲✶✻✺✵✮✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣✉r♣♦s❡ ✐s t♦ r❡❜✉✐❧❞ ✐♠♣♦rt❛♥t ♣♦✐♥ts ♦❢ t❤❡ s✉❜❥❡❝t ❢r♦♠ ▼❡s♦♣♦t❛♠✐❛♥s ❛♥❞ ❊❣②♣t✐❛♥s t♦ t❤❡ ♣r❡s❡♥t t✐♠❡✳ ❘❡♠❛r❦❛❜❧②✱ ❊✉❝❧✐❞✬s ❊❧❡♠❡♥ts ✭✸✵✵ ❇✳❈✳✮ s❤♦✇s s♦♠❡ ♣r♦♣♦s✐t✐♦♥s ❛❜♦✉t t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❛r❡❛s ❛♥❞ ❛❜♦✉t t❤❡ ❣♦❧❞❡♥ s❡❝t✐♦♥✱ ✇❤✐❝❤ ❜❡❤❛✈❡ ❧✐❦❡ t②♣✐❝❛❧ ❝❛s❡s ♦❢ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ■t ✐s ❛❧s♦ ♦❜s❡r✈❡❞ t❤❡ ❣r❡❛t ❝♦♥tr✐❜✉t✐♦♥ ♦❢ ❍✐♥❞✉s ❛♥❞ ❆r❛❜s✱ ✇❤♦ ✐♥tr♦❞✉❝❡❞✱ t❤r♦✉❣❤ r❡❝✐♣❡s ❛♥❞ ❣❡♦♠❡tr✐❝ ❢♦r♠s✱ t❤❡ ✐♠♣♦rt❛♥t ♠❡t❤♦❞ ♦❢ ❝♦♠♣❧❡t✐♥❣ sq✉❛r❡s✱ ✇❤✐❝❤ ✇❛s ❢✉♥❞❛♠❡♥t❛❧ t♦ r❡❛❝❤ t❤❡ ❝❧❛ss✐❝ ❢♦r♠✉❧❛✳ ❋✐♥❛❧❧②✱ t❤❡ s♦✲ ❧✉t✐♦♥s ♦❢ ❱✐èt❡ ❛♥❞ ❉❡s❝❛rt❡s ❝❛♥ ❜❡ ❝❛❧❧❡❞ ❛❧❣❡❜r❛✐❝ ❛♥❞ ❛♥❛❧②t✐❝ s♦❧✉t✐♦♥s r❡s♣❡❝t✐✈❡❧②✳ ❙✐♠✐❧❛r❧② t♦ t♦❞❛② ❢♦r♠✉❧❛✱ ✐♥ ❜♦t❤ ❝❛s❡s✱ s♣❡❝✐✜❝❛❧❧②✱ ❧❡tt❡rs ✇❡r❡ ✉s❡❞ t♦ r❡♣r❡s❡♥t t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥s✳
❑❡②✇♦r❞s✿ ❍✐st♦r② ♦❢ ❆❧❣❡❜r❛✱ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳
✏❙ó ♥❛ ❢♦③ ❞♦ r✐♦ é q✉❡ s❡ ♦✉✈❡♠ ♦s ♠✉r♠úr✐♦s ❞❡ t♦❞❛s ❛s ❢♦♥t❡s✑ ✭❏♦ã♦ ●✉✐♠❛rã❡s ❘♦s❛✮
❊●■❚❖
❙ã♦ ❝♦♥❤❡❝✐❞♦s ♣♦✉❝♦s r❡❣✐str♦s ❞♦ tr❛t❛♠❡♥t♦ ❞❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉ ♣❡❧♦s ❡❣í♣❝✐♦s✱ ♠❛s ♦s ❤✐st♦r✐❛❞♦r❡s s✉s♣❡✐t❛♠ q✉❡ ❡❧❡s ❞♦♠✐♥❛✈❛♠ ❛❧❣✉♠❛ té❝♥✐❝❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ss❛s ❡q✉❛çõ❡s✳ ❯♠ ❡①❡♠♣❧♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ P❛♣✐r♦ ❞❡ ❇❡r❧✐♠ ❡ r❡♠♦♥t❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛♦ ❛♥♦ ✶✾✺✵ ❛✳❈✳ ❚❛♠❜é♠ ❢♦✐ ❡♥❝♦♥tr❛❞❛ ♥♦ P❛♣✐r♦ ❞❡ ❑❛❤✉♥ ✉♠❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦✱ ❤♦❥❡ ❡s❝r✐t❛ ❝♦♠♦
x2
+y2
=k✱k✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦✱ ♣❡❧♦ ♠ét♦❞♦ ❞❛ ❢❛❧s❛ ♣♦s✐çã♦✱ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧♦s ❡❣í♣❝✐♦s
♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ ✶➦ ❣r❛✉✳
❊①❡♠♣❧♦✿ ❆ s♦♠❛ ❞❛s ár❡❛s ❞❡ ❞♦✐s q✉❛❞r❛❞♦s é ✶✵✵ ✉♥✐❞❛❞❡s✳ ❖ tr✐♣❧♦ ❞♦ ❧❛❞♦ ❞❡ ✉♠ ❞❡❧❡s é ♦ q✉á❞r✉♣❧♦ ❞♦ ❧❛❞♦ ❞♦ ♦✉tr♦✳ ❊♥❝♦♥tr❡ ♦s ❧❛❞♦s ❞❡ss❡ q✉❛❞r❛❞♦✳
❊♠ s✐♠❜♦❧♦❣✐❛ ❛t✉❛❧ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s q✉❡ r❡♣r❡s❡♥t❛ ♦ ♣r♦❜❧❡♠❛ é
(
x2
+y2
= 100
y = 4 3x
❆ s❡❣✉✐r ♦ ♣r♦❝❡❞✐♠❡♥t♦ r❡tór✐❝♦ ❞❛❞♦ ♣❡❧♦ ❡s❝r✐❜❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✿ ✶✳ ❚♦♠❡ x= 3, ❡♥tã♦✱ y= 4
✷✳ ❆ss✐♠✱ 32
+ 42
= 25·(25 6= 100)
✸✳ √25 = 5, √100 = 10
✹✳ 10÷5 = 2
✺✳ ❖s ❧❛❞♦s sã♦ 2×3 = 6 ❡ 2×4 = 8. ✭P❛♣✐r♦ ❞❡ ❇❡r❧✐♠✮
▼❊❙❖P❖❚➶▼■❆
❖ ♣r✐♠❡✐r♦ r❡❣✐str♦ ❝♦♥❤❡❝✐❞♦ ❞❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉ ❞❛t❛ ❞❡ ✶✼✵✵ ❛✳❈✳ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ❢❡✐t♦ ♥✉♠❛ tá❜✉❛ ❞❡ ❛r❣✐❧❛ ❛tr❛✈és ❞❡ ♣❛❧❛✈r❛s✳ ❆ s♦❧✉çã♦ ❡r❛ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ ✉♠❛ ✏r❡❝❡✐t❛ ♠❛t❡♠át✐❝❛✑ ❡ ❢♦r♥❡❝✐❛ s♦♠❡♥t❡ ✉♠❛ r❛✐③ ♣♦s✐t✐✈❛✳ ❖s ♠❡s♦♣♦tâ♠✐♦s ❡♥✉♥❝✐❛✈❛♠ ❛ ❡q✉❛çã♦ ❡ s✉❛ r❡s♦❧✉çã♦ ❡♠ ♣❛❧❛✈r❛s✱ ♠❛✐s ♦✉ ♠❡♥♦s ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
❊①❡♠♣❧♦✿ ◗✉❛❧ é ♦ ❧❛❞♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡♠ q✉❡ ❛ ár❡❛ ♠❡♥♦s ♦ ❧❛❞♦ ❞á ✽✼✵❄ ✭❖ q✉❡ ❤♦❥❡ s❡ ❡s❝r❡✈❡✿ x2
−x= 870✮✳ ❊ ❛ ✏r❡❝❡✐t❛✑ ❡r❛✿
❚♦♠❡ ❛ ♠❡t❛❞❡ ❞❡ ✶ ✭❝♦❡✜❝✐❡♥t❡ ❞❡ x✮ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣♦r ❡❧❛ ♠❡s♠❛✱ (0,5×0,5 = 0,25)✳
❙♦♠❡ ♦ r❡s✉❧t❛❞♦ ❛ 870 ✭t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡✮✳ ❖❜té♠✲s❡ ✉♠ q✉❛❞r❛❞♦ (870,25 = (29,5)2
)
❝✉❥♦ ❧❛❞♦ s♦♠❛❞♦ à ♠❡t❛❞❡ ❞❡1 ✈❛✐ ❞❛r (30)✱ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ♣r♦❝✉r❛❞♦✳
●❘➱❈■❆
❊✉❝❧✐❞❡s ✕ ❞❡t❛❧❤❡ ❞❡ ❆ ❊s❝♦❧❛ ❞❡ ❆t❡♥❛s ❞❡ ❘❛❢❛❡❧
❆❝r❡❞✐t❛✲s❡ q✉❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ♥♦ tr❛t❛♠❡♥t♦ ❝♦♠ ♦s ♥ú♠❡r♦s✱ r❛❝✐♦♥❛✐s ❡ ✐rr❛❝✐♦♥❛✐s✱ ❡ ❛ ❢❛❧t❛ ❞❡ ♣r❛t✐❝✐✲ ❞❛❞❡ ❞♦ s✐st❡♠❛ ❞❡ ♥✉♠❡r❛çã♦ ❣r❡❣♦✱ q✉❡ ❡r❛ ❧✐t❡r❛❧✱ ❛❧é♠ ❞♦ ❣♦st♦ ♥❛t✉r❛❧ ♣❡❧❛ ❣❡♦♠❡tr✐❛✱ ❧❡✈♦✉ ❡ss❛ ❝✐✈✐✲ ❧✐③❛çã♦ ✭✺✵✵ ❛ ✷✵✵ ❛✳❈✳✮ ❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ tr❛t❛♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❞❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s✱ ❞❡♥tr❡ ♦s q✉❛✐s✱ ❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ ✷➦ ❣r❛✉✳
❊♠ ✏❖s ❊❧❡♠❡♥t♦s✑ ❞❡ ❊✉❝❧✐❞❡s ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♥❝♦♥tr❛✲s❡ ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s ❞❡ss❡ t✐♣♦ ❞❡ ❡q✉❛çã♦✳
Pr♦♣♦s✐çã♦ ✷✽ ✕ ▲✐✈r♦ ❱■✿ ❉✐✈✐❞✐r ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡ ♠♦❞♦ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❝♦♥t✐❞♦ ♣♦r s✉❛s ♣❛rt❡s s❡❥❛ ✐❣✉❛❧ ❛ ✉♠ q✉❛❞r❛❞♦ ❞❛❞♦✱ ♥ã♦ ❡①❝❡❞❡♥❞♦ ❡st❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ♠❡t❛❞❡ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❛❞❛✳ ❊♠ ❧✐♥❣✉❛❣❡♠ ❛t✉❛❧✱ x2
−px+q2
= 0✱ ❡♠ q✉❡ p ❡ q sã♦
s❡❣♠❡♥t♦s ❞❛❞♦s✳
q
❆ P ◗ ❇
❊
q
❙❡❥❛♠ AB ❡ P E ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛✱ ❡♠ q✉❡ AB =p✱ P E = q ❡ q < p
2✳ ❉✐✈✐❞✐♥❞♦
AB ❝♦♠ ♦ ♣♦♥t♦ Qt❛❧ q✉❡AQ+QB =p ❡AQ·QB =q2
t❡♠✲s❡ ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛✳ P❛r❛ ✐ss♦ ❜❛st❛ tr❛ç❛r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ ❝❡♥tr♦ ❡♠E ❡ r❛✐♦ p
2✱ q✉❡ ❝♦rt❛rá s❡❣♠❡♥t♦ AB ♥♦
♣♦♥t♦Q✳ ▲♦❣♦
q2
=P B2−P Q2 = (P B−P Q)·(P B+P Q) =QB ·AQ.
❋✐♥❛❧♠❡♥t❡ ❞❡♥♦t❛♥❞♦ ♣♦r r = AQ ❡ s = BQ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞❛❞❛✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ p=r+s ❡ q2
=r·s✳
❊①❡♠♣❧♦✿ x2
−3x+ 1 = 0.
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♥♦t❛çõ❡s ❛♥t❡r✐♦r❡s t❡♠✲s❡ p = 3 ❡ q = 1✳ P♦r ❝♦♥str✉çã♦✱ EQ =
P B = 3
2 ❡ P E = q = 1✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ s❡q✉❡ q✉❡ P Q =
√
5
2 ❡✱ ❛ss✐♠✱
s = AQ = AP +P Q = 3 2+
√
5
2 . ❆❣♦r❛✱ r = QB = AB −AQ = 3−
3 +√5
2 ✱ ♦✉ s❡❥❛✱
r= 3 +
√
5
2 ✳
Pr♦♣♦s✐çã♦ ✷✾ ✕ ▲✐✈r♦ ❱■ ✿ Pr♦❧♦♥❣❛r ✉♠ ❞❛❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡ ♠♦❞♦ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❝♦♥t✐❞♦ ♣❡❧♦ s❡❣♠❡♥t♦ ❡st❡♥❞✐❞♦ ❡ ❛ ❡①t❡♥sã♦ s❡❥❛ ✐❣✉❛❧ ❛ ✉♠ q✉❛❞r❛❞♦ ❞❛❞♦✳ ❊♠ ❧✐♥❣✉❛❣❡♠ ❛t✉❛❧✱x2
−px−q2
= 0✳
q
❆ P ❇
❊
q
❙❡❥❛♠ AB ❡ BE ❡ ❞♦✐s s❡❣♠❡♥t♦s ❞❡ r❡t❛✱ ❡♠ q✉❡ AB = p ❡ BE = q✳ ❉❡t❡r♠✐♥❛✲s❡ ♦
♣♦♥t♦Q t❛❧ q✉❡ AQ+QB =p ❡AQ·QB =q2
✳ P❡❧❛ ❝♦♥str✉çã♦ t❡♠✲s❡ P E =P Q✱ ❡♠ q✉❡ P é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ AB✳ ◆♦t❛✲s❡ q✉❡q2
+P B2 =P E2 =P Q2✱ ♦✉ s❡❥❛✱
q2
=P Q2−P B2 = (P Q−P B)·(P Q+P B) =QB·(P Q+P A) = QB·AQ.
❈♦♥s✐❞❡r❛♥❞♦ r =AQ ❡ s = QB✱ ❡♥tã♦ p= r−s ❡ r·(−s) = −rs= −QB·AQ =−q2
❡✱ ❛ss✐♠✱r ❡ ✕s sã♦ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞❛❞❛✳
❊①❡♠♣❧♦✿ x2
−13x−9 = 0✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛s ♥♦t❛çõ❡s ❛♥t❡r✐♦r❡s t❡♠✲s❡ p = 13 ❡ q = 3✳ P♦r ❝♦♥str✉çã♦✱ AP =
P B = 13
2 ❡ BE = q = 3✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ s❡❣✉❡ q✉❡ P E =
√
205
2 ✳ ❈♦♠♦✱
P E = P Q = P B +BQ = P B −QB✱ ❡♥tã♦ s = QB = P B −P E = 13 2 +
√
205 2 ▲♦❣♦✱
s = 13 +
√
205
2 . ❆❣♦r❛✱ r = AQ = AB +BQ = AB −QB = 13 −
13 +√205
2 ✱ ✐st♦ é✱
r= 13−
√
205
2 .
Pr♦♣♦s✐çã♦ ✶✶ ✕▲✐✈r♦ ■■ ✭❙❡❣♠❡♥t♦ ➪✉r❡♦✮✿ ❉✐✈✐❞✐r ✉♠❛ ❧✐♥❤❛ r❡t❛ ❡♠ ❞✉❛s ♣❛rt❡s t❛✐s q✉❡ ♦ r❡tâ♥❣✉❧♦ ❝♦♥t✐❞♦ ♣❡❧♦ t♦❞♦ ❡ ✉♠❛ ❞❛s ♣❛rt❡s t❡♥❤❛ ár❡❛ ✐❣✉❛❧ à ❞♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ♦✉tr❛ ♣❛rt❡✳ ❉❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡✱ ❞❛❞♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ AB✱ ❞❡✈❡✲s❡ ❞❡t❡r♠✐♥❛r
♦ ♣♦♥t♦ X ❞❡ss❡ s❡❣♠❡♥t♦ t❛❧ q✉❡ ♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s AB ❡ XB t❡♥❤❛ ❛ ♠❡s♠❛ ár❡❛ ❞♦
q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ AX✳
■♥❞✐❝❛♥❞♦✲s❡ ❛s ♠❡❞✐❞❛s ❞❡ AB ❡ AX ♣♦r a ❡ x✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♠♦str❛✲s❡ q✉❡ a ❡ x
❞❡✈❡♠ s❛t✐s❢❛③❡r ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦✿ a(a−x) = x2
✳
◆✉♠❛ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✱ ❡ ❡♠ ♥♦t❛çã♦ ❛t✉❛❧✱ ❛ s♦❧✉çã♦ ❞❡ ❊✉❝❧✐❞❡s ❝♦♠♣õ❡✲s❡ ❞♦s s❡❣✉✐♥t❡s ♣❛ss♦s✿
✶✳ ❈♦♥str✉✐r ♦ q✉❛❞r❛❞♦ ABCD s♦❜r❡ ♦ s❡❣♠❡♥t♦ ❞❛❞♦ AB❀
✷✳ ❚♦♠❛r ♦ ♣♦♥t♦ ♠é❞✐♦✱E✱ ❞❡ DA❀
✸✳ ❚♦♠❛r F s♦❜r❡ ♦ ♣r♦❧♦♥❣❛♠❡♥t♦ ❞❡ DA ❞❡ ♠❛♥❡✐r❛ q✉❡EF =EB❀
✹✳ ❈♦♥str✉✐r ♦ q✉❛❞r❛❞♦ s♦❜r❡ ♦ ❧❛❞♦AF ♥♦ ♠❡s♠♦ s❡♠✐✲♣❧❛♥♦ ❞❡BC✳
✺✳ ❖ ✈ért✐❝❡ X ❞❡ss❡ q✉❛❞r❛❞♦✱ ♣❡rt❡♥❝❡♥t❡ ❛♦ s❡❣♠❡♥t♦ AB✱ é ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳
❉❡ ❢❛t♦✿
AE = 1 2AD=
1
2a. P♦rt❛♥t♦✱ ♥♦ tr✐â♥❣✉❧♦ABE t❡♠✲s❡EB =
r
a2
+a 2
2
= a
√
5 2 ✳ ❉❛í✱
x = AX = AF = EF −EA = EB −EA = a
√
5
2 −
a
2 =
a(−1 +√5)
2 é ❛ r❛✐③ ♣♦s✐t✐✈❛ ❞❡
a(a−x) =x2
, ❞❡♥♦♠✐♥❛❞♦ ♥ú♠❡r♦ á✉r❡♦✱ q✉❡ é ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦AX.
❉ ❈
❊ ❛
❆ ① ❳ ❇
❋ ●
❮◆❉■❆
❆ ♠❛t❡♠át✐❝❛ ❤✐♥❞✉ ♣r♦❞✉③✐✉ ❛té ♦ r❡♥❛s❝✐♠❡♥t♦ ❣r❛♥❞❡s ♣❡rs♦♥❛❣❡♥s✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝❛♠✲s❡ ❆r②❛❜❤❛t❛ ✭sé❝✳ ❱■ ❞✳❈✳✮✱ ❇r❛❤❛♠❛❣✉♣t❛ ✭sé❝✳ ❱■■ ❞✳❈✳✮✱ ❙r✐❞❤❛r❛ ✭sé❝✳ ❳■ ❞✳❈✳✮ ❡ ❇❤❛s❦❛r❛ ✭✶✶✶✹✲✶✶✽✺✮✱ q✉❡ ♠✉✐t♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ✷➸ ❣r❛✉ ❛♦ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s✳ ❙❡❣✉♥❞♦ ♦ ♣ró♣r✐♦ ❇❤❛s❦❛r❛ ❛ r❡❣r❛ q✉❡ ✉s❛✈❛ ❡ q✉❡ ♦r✐❣✐♥♦✉ ❛ ❢ór♠✉❧❛ ❛t✉❛❧ ❡r❛ ❞❡✈✐❞♦ ❛ ❙r✐❞❤❛r❛ ❡ q✉❡ ❝✉r✐♦s❛♠❡♥t❡ é ❝❤❛♠❛❞❛✱ s♦♠❡♥t❡ ♥♦ ❇r❛s✐❧✱ ❞❡ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✳
◆♦t❛çõ❡s ❤✐♥❞✉s ♣❛r❛ ❛s ❡q✉❛çõ❡s✿
②❛ ✭❛❜r❡✈✐❛çã♦ ❞❡ ②❛✈❛tt❛✈❛t✮ ❡r❛ ❛ ♣r✐♠❡✐r❛ ✐♥❝ó❣♥✐t❛❀ ❦❛ ✭ ❦❛❧❛❦❛ ♦✉ ✏♥❡❣r♦✑✮ ❡r❛ ❛ s❡❣✉♥❞❛ ✐♥❝ó❣♥✐t❛❀
✈ ✭✈❛r❣❛✮ s✐❣♥✐✜❝❛✈❛ ✏q✉❛❞r❛❞♦✑❀
˙ ❯♠ ♣♦♥t♦ s♦❜r❡ ♦ ♥ú♠❡r♦ ✐♥❞✐❝❛✈❛ q✉❡ ❡❧❡ ❡r❛ ♥❡❣❛t✐✈♦❀
❜❤❛ ✭❜❤❛✈✐t❛✮ s✐❣♥✐✜❝❛✈❛ ✏♣r♦❞✉t♦✑❀
❦✭❛✮ r❡♣r❡s❡♥t❛✈❛ ❦❛r❛♥❛ ✭✏✐rr❛❝✐♦♥❛❧✑ ♦✉ ✏r❛✐③✑✮❀ r✉ r❡♣r❡s❡♥t❛✈❛ r✉♣❛ ✭♥ú♠❡r♦ ✏♣✉r♦✑ ♦✉ ✏❝♦♠✉♠✑✮✳
❖ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ❡q✉❛çã♦ ❡r❛ ❡s❝r✐t♦ ❡♠ ✉♠❛ ❧✐♥❤❛ ❡ ♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ♥❛ ❧✐♥❤❛ ❛❜❛✐①♦❀
■♥❝ó❣♥✐t❛s ❛❞✐❝✐♦♥❛✐s s❡r✐❛♠ ❡①♣r❡ss❛s ♠❡❞✐❛♥t❡ ♦ ✉s♦ ❞❡ ❛❜r❡✈✐❛çõ❡s ♣❛r❛ ❝♦r❡s ❛❞✐❝✐♦♥❛✐s✱ ❛ss✐♠✿ ♥✐ ♣❛r❛ ♥✐❧❛❝❛ ✭✏❛③✉❧✑✮✱ ♣✐ ♣❛r❛ ♣✐t❛❝❛ ✭✏❛♠❛r❡❧♦✑✮✱ ♣❛ ♣❛r❛ ♣❛♥❞✉ ✭✏❜r❛♥❝♦✑✮ ❡ ❧♦ ♣❛r❛ ❧♦❤✐t❛ ✭✏✈❡r♠❡❧❤♦✑✮✳
❊①❡♠♣❧♦ ✶✿ ya vru1ya˙9 10˙ tr❛❞✉③✲s❡ ♣♦rx2
−10x=−9✳
❆ s❡❣✉✐r✱ ♦s ♣❛ss♦s ❞❛ s♦❧✉çã♦ ❞❡ ❇r❛❤❛♠❛❣✉♣t❛✱ ❡♠ ♥♦t❛çã♦ ❛t✉❛❧✿ ❼ −9
❼ (−9)·4 =−36
❼ −36 + (−10)2
= 64
❼ √64 = 8
❼ 8−(−10) = 18
❼ 18 : (2×1) = 9
❼ ❆ r❛✐③ é9.
❊①❡♠♣❧♦ ✷✿ ya kaya v7bha k3ya(a) 1210 ru ˙8 tr❛❞✉③✲s❡ ♣♦r 7xy+ √
12−8 = = 3x2
+ 10x
❇❤❛s❦❛r❛ ✭✶✶✶✹ ✕ ✶✶✽✺✮✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏♦ sá❜✐♦✑ ✢♦r❡s❝❡✉ ❝✐♥❝♦ sé❝✉❧♦s ❞❡♣♦✐s ❞❡ ❇r❛❤✲ ♠❛❣✉♣t❛✳ ▼❛t❡♠át✐❝♦✱ ♣r♦❢❡ss♦r✱ ❛stró❧♦❣♦ ❡ ❛strô♥♦♠♦✱ ♣r❡❡♥❝❤❡✉ ❧❛❝✉♥❛s ❞❡✐①❛❞❛s ♣♦r s❡✉s ❛♥t❡❝❡ss♦r❡s✱ ✐♥❝❧✉s✐✈❡✱ ❞❛♥❞♦ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦x2
= 1 +py2❡ ❞❡ ♠✉✐t❛s ♦✉tr❛s
❡q✉❛çõ❡s ❞✐♦❢❛♥t✐♥❛s✳ ❋❡③ ♥♦tá✈❡✐s ♣r♦❣r❡ss♦s ♥❛ ♥♦t❛çã♦ ❛❧❣é❜r✐❝❛ ❛❜r❡✈✐❛❞❛ ❡ ✐ss♦ ♣♦❞❡ s❡r ❝♦♥✜r♠❛❞♦ ♥❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✷✱ ❛ s❡❣✉✐r✳
❉♦s s❡✉s s❡✐s tr❛❜❛❧❤♦s ❝♦♥❤❡❝✐❞♦s ♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s sã♦ ▲✐❧❛✈❛t✐ ✭♥♦♠❡ ❞❡ s✉❛ ✜❧❤❛ ❡ q✉❡ ❝♦♥té♠ ✷✼✽ ✈❡rs♦s✮ ❡ ❱✐❥❛✲●❛♥✐t❛✱ ❛♠❜♦s ❝♦♠ ♠✉✐t♦s ♣r♦❜❧❡♠❛s s♦❜r❡ ♦s tó♣✐❝♦s ❢❛✈♦r✐t♦s ❞♦s ❤✐♥❞✉s✿ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❡ q✉❛❞rát✐❝❛s ✭❞❡t❡r♠✐♥❛❞❛s ♦✉ ✐♥❞❡t❡r♠✐♥❛❞❛s✮✱ ♠❡♥s✉r❛çã♦✱ ♣r♦❣r❡ssõ❡s ❛r✐t♠ét✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✱ r❛❞✐❝❛✐s✱ t❡r♥❛s ♣✐t❛❣ór✐❝❛s✱ r❡❣r❛ ❞❡ três✱ ❡t❝✳
❊①❡♠♣❧♦s ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ▲✐❧❛✈❛t✐✿
Pr♦❜❧❡♠❛ ✶✿ ❖ q✉❛❞r❛❞♦ ❞❛ q✉✐♥t❛ ♣❛rt❡ ❞♦ ♥ú♠❡r♦ ❞❡ ♠❛❝❛❝♦s ❞❡ ✉♠ ❜❛♥❞♦✱ s✉❜tr❛í❞❛ ❞❡ ✸ ♠❛❝❛❝♦s✱ ❡♥tr❛ ♥✉♠❛ ❝❛✈❡r♥❛❀ ❡ ✉♠ ♠❛❝❛❝♦ ✜❝❛ ❢♦r❛ ♣❡♥❞✉r❛❞♦ ♥✉♠❛ ár✈♦r❡✳ ❉✐❣❛ q✉❛♥t♦s sã♦ ♦s ♠❛❝❛❝♦s✳
❊♠ ♥♦t❛çã♦ ❛t✉❛❧ t❡♠✲s❡✿
1 5x−3
2
+ 1 =x♦✉ x2
−55 =−250✳
Pr♦❜❧❡♠❛ ✷✿ ❆ r❛✐③ q✉❛❞r❛❞❛ ❞♦ ♥ú♠❡r♦ ❞❡ ❛❜❡❧❤❛s ❞❡ ✉♠ ❡♥①❛♠❡ ✈♦♦✉ r✉♠♦ ❛ ✉♠ ❥❛s♠✐♥❡✐r♦✱ ❡♥q✉❛♥t♦ 8/9 ❞♦ ❡♥①❛♠❡ ♣❡r♠❛♥❡❝❡✉ ❛trás❀ ❡ ✉♠❛ ❛❜❡❧❤❛ ❢ê♠❡❛ ✜❝♦✉ ✈♦❛♥❞♦ ❡♠
t♦r♥♦ ❞❡ ✉♠ ♠❛❝❤♦ q✉❡ s❡ ❡♥❝♦♥tr❛✈❛ ♣r❡s♦ ♥✉♠❛ ✢♦r ❞❡ ❧ót✉s ♣❛r❛ ❛ q✉❛❧ ❢♦✐ ❛tr❛í❞♦ à ♥♦✐t❡ ♣♦r s❡✉ ❞♦❝❡ ♦❞♦r✳ ❉✐❣❛✲♠❡ ❛❞♦rá✈❡❧ ♠✉❧❤❡r✱ q✉❛❧ é ♦ ♥ú♠❡r♦ ❞❡ ❛❜❡❧❤❛s✳
◆❛ t❛❜❡❧❛ q✉❡ s❡❣✉❡✱ ♥❛ ❝♦❧✉♥❛ ❞❛ ❡sq✉❡r❞❛ t❡♠✲s❡ ❛ s♦❧✉çã♦ ❞❡ ❇❤❛s❦❛r❛ ❡ ♥❛ ❞❛ ❞✐r❡✐t❛ ❛ tr❛❞✉çã♦ ❛t✉❛❧✳
❙❡❥❛ ya v 2♦ ♥ú♠❡r♦ ❞❡ ❛❜❡❧❤❛s ❞♦
❡♥①❛♠❡ ❙❡❥❛ 2x
2♦ ♥ú♠❡r♦ ❞❡ ❛❜❡❧❤❛s ❞♦
❡♥①❛♠❡ ❆ r❛✐③ q✉❛❞r❛❞❛ ❞❛ ♠❡t❛❞❡ ❞❡ss❡
♥ú♠❡r♦ éya 1
r
2x2
2 =x
❖✐t♦ ♥♦♥♦s ❞❡ t♦❞♦ ♦ ❡♥①❛♠❡ é
ya v 16
9
❖✐t♦ ♥♦♥♦s ❞❡ t♦❞♦ ♦ ❡♥①❛♠❡ é
16 9
x2
❆ s♦♠❛ ❞❛ r❛✐③ q✉❛❞r❛❞❛ ❝♦♠ ❛ ❢r❛çã♦ ❡ ♦ ❝❛s❛❧ ❞❡ ❛❜❡❧❤❛s é ✐❣✉❛❧ à
q✉❛♥t✐❞❛❞❡ ❞❡ ❛❜❡❧❤❛s ❞♦ ❡♥①❛♠❡✱ ✐st♦ é✱ya v 2
x+
16 9
x2
+ 2 = 2x2
❞❡♥♦♠✐♥❛❞♦r ♦s ❞♦✐s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ❡ ❡❧✐♠✐♥❛♥❞♦ ♦ ❞❡♥♦♠✐♥❛❞♦r✱ ❛ ❡q✉❛çã♦
tr❛♥s❢♦r♠❛✲s❡ ❡♠
ya v 18 ya 0 ru 0
ya v 16 ya 9 ru 18
9x+ 16x2
+ 18
9 =
18x2
9 ⇐⇒
18x2
= 16x2
+ 9x+ 18
❆♣ós ❛ s✉❜tr❛çã♦ ❛ ❡q✉❛çã♦ t♦r♥❛✲s❡
ya v 2 ya 9 ru 0
ya v 0 ya ru18 18x2
−16x2
−9x = 16x2
+ 9x+ 18−16x2
−9x
2x2
−9x = 18
P♦rt❛♥t♦ ya é ✻ P♦rt❛♥t♦ x= 6
❉♦♥❞❡ ya v 2 é72 ❉♦♥❞❡ 2x2
= 2·62
= 72
▼❡r❡❝❡ ❞❡st❛q✉❡✱ ❛ tít✉❧♦ ❞❡ ❝✉r✐♦s✐❞❛❞❡✱ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s r❡❛❧✐③❛❞❛ ♣♦r ❇❤❛s❦❛r❛✿ ❡s❜♦ç♦✉ ❛ ✜❣✉r❛ ❛❜❛✐①♦ ❡ ❡s❝r❡✈❡✉ s✐♠♣❧❡s♠❡♥t❡ ✏❱❡❥❛✑✳
❆❘➪❇■❆
❙❡✱ ♣♦r ✉♠ ❧❛❞♦✱ ❝♦♠♦ ❞✐③ ❛ tr❛❞✐çã♦✱ ♦s ár❛❜❡s ❢♦r❛♠ r❡s♣♦♥sá✈❡✐s ♣❡❧♦ ❞❡s❛♣❛r❡❝✐♠❡♥t♦ ❞♦ s❛❜❡r ♦❝✐❞❡♥t❛❧✱ ♣♦r ♦✉tr♦ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ s✉❛ ♣r❡s❡r✈❛çã♦✳ ❙❡❣✉♥❞♦ ❝♦♥st❛✱ ♦ ❡①t❡r✲ ♠í♥✐♦ s❡ ❞❡✉ q✉❛♥❞♦✱ ❡♠ ✻✹✶ ❞✳❈✳✱ ♦ ❝❛❧✐❢❛ ❖♠❛r ♠❛♥❞♦✉ q✉❡ ❢♦ss❡ ❞❡str✉í❞❛ ❛ ❇✐❜❧✐♦t❡❝❛ ❞❡ ❆❧❡①❛♥❞r✐❛✳ ❊ ❛ ♣r❡s❡r✈❛çã♦ ❢♦✐ ♦❜r❛ ❞❡ três ❝❛❧✐❢❛s✱ ❝♦♥s✐❞❡r❛❞♦s ♦s ❣r❛♥❞❡s ♣❛tr♦♥♦s ❞❛ ❝✉❧t✉r❛ ❛❜áss✐❞❛✿ ❛❧✲▼❛♥s✉r✱ ❍❛r✉♠ ❛❧✲❘❛❝❤✐❞ ❡ ❛❧✲▼❛♠✉♠✱ q✉❡ ❞✉r❛♥t❡ s❡✉s r❡✐♥❛❞♦s ❢♦r❛♠ r❡s♣♦♥sá✈❡✐s ♣❡❧❛ tr❛❞✉çã♦✱ ❞♦ ❣r❡❣♦ ♣❛r❛ ♦ ár❛❜❡✱ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❡s❝r✐t♦s ❝✐❡♥tí✜❝♦s ❝♦♥❤❡❝✐❞♦s✱ ❡♥tr❡ ❡❧❡s✱ ❖ ❆❧♠❛❣❡st♦ ❞❡ Pt♦❧♦♠❡✉ ❡ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✳
❆❧✲▼❛♠✉♠ ❢✉♥❞♦✉ ❡♠ ❇❛❣❞á✱ ♥♦ sé❝✉❧♦ ■❳✱ ✉♠ ❝❡♥tr♦ ❝✐❡♥tí✜❝♦ s✐♠✐❧❛r à ❇✐❜❧✐♦t❡❝❛ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❞❡♥♦♠✐♥❛❞♦ ❈❛s❛ ❞❛ s❛❜❡❞♦r✐❛ ✭❇❛✐t ❛❧✲❤✐❦♠❛✮✱ ♣❛r❛ ♦♥❞❡ ❝♦♥✈❡r❣✐r❛♠ ♠✉✐t♦s
♠❛t❡♠át✐❝♦s✱ ❞❡♥tr❡ ♦s q✉❛✐s ▼♦❤❛♠❡❞ ✐❜♥✲▼✉s❛ ❛❧✲❑❤♦✇❛r✐③♠✐✱ q✉❡✱ ❛❧é♠ ❞❡ ♦✉tr❛s ♦❜r❛s✱ ❡s❝r❡✈❡✉✱ ❡♠ ✽✷✺✱ ❍✐s❛❜ ❛❧✲❥❛❜r ✇❛✬❧♠✉q❛❜❛❧❛❤ ✭❝✐ê♥❝✐❛ ❞❛ r❡st❛✉r❛çã♦ ❡ ❞❛ r❡❞✉çã♦ ♦✉ ❝✐ê♥✲ ❝✐❛ ❞❛s ❡q✉❛çõ❡s✮✱ ♦❜r❛ ❞❡ ❣r❛♥❞❡ ♣♦t❡♥❝✐❛❧ ❞✐❞át✐❝♦✳ ◆❡ss❛ ♦❜r❛✱ ❛❧✲❑❤♦✇❛r✐③♠✐ ❛♣r❡s❡♥t❛ ❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉✱ ❜❡♠ ❝♦♠♦ s✉❛ r❡s♦❧✉çã♦✱ ❞❡ ❢♦r♠❛ r❡tór✐❝❛✱ ❛❧é♠ ❞❡ ✉♠❛ ❝♦♠♣r♦✈❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡♥♦♠✐♥❛❞❛ ♠ét♦❞♦ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦s✱ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ❞✐st✐♥t♦ ❞♦ ✉t✐✲ ❧✐③❛❞♦ ♣❡❧♦s ❣r❡❣♦s✳ ❊♠ ♠✉✐t♦s ❝❛s♦s ❛♣r❡s❡♥t❛✈❛✱ t❛❧ ❝♦♠♦ s❡✉s ♣r❡❞❡❝❡ss♦r❡s s♦♠❡♥t❡ ✉♠❛ r❛✐③ ✭♣♦s✐t✐✈❛✮✳
❊①❡♠♣❧♦✿ ❊♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ♣❛r❛ x2
+ 12x= 64.
❊ ● ❋
✻
❑ ❇ ❈
❍ ✻ ❆ x ❉
x
◆❛ ✜❣✉r❛ ❛♦ ❧❛❞♦ t❡♠✲s❡ q✉❡ AB =BC =x ❡ q✉❡ AH =CF = 6. ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛
ár❡❛ ❞♦ q✉❛❞r❛❞♦ ABCD é ❞❛❞❛ ♣♦r Aq = x
2
❡ ❛ ár❡❛ ❞♦s r❡tâ♥❣✉❧♦s HKBA ❡ BGF C é
❞❛❞❛ ♣♦rAr = 6x. ❆ s♦♠❛ ❞❡ss❛s ár❡❛s é x
2
+ 6x+ 6x=x2
+ 12x.❈♦♠♣❧❡t❛✲s❡ ♦ q✉❛❞r❛❞♦ HEF D ❝♦♠ ♦ q✉❛❞r❛❞♦KEGB, ❝✉❥❛ ár❡❛ é ❞❛❞❛ ♣♦rA′
q = 36.❆ ár❡❛ ❞♦ q✉❛❞r❛❞♦HEF D
é ❞❛❞❛ ♣♦r (x+ 6)2
=x2
+ 12x+ 36 = 64 + 36 = 100, ♦ q✉❡ r❡s✉❧t❛ x= 4.
❆❧✲❑❤♦✇❛r✐③♠✐ só ❝♦♥s✐❞❡r❛✈❛ ❛s r❛í③❡s ♣♦s✐t✐✈❛s✱ ♠❛s✱ ❛♦ ❝♦♥trár✐♦ ❞♦s ❣r❡❣♦s✱ ❛❞♠✐t✐❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞✉❛s r❛í③❡s✳
❈❍■◆❆
❊♠ ✶✸✵✸✱ ♦ ❣r❛♥❞❡ ♠❛t❡♠át✐❝♦ ❝❤✐♥ês✱ ❈❤✉ ❙❤✐❤✲❝❤✐❡❤✱ ❛♣r❡s❡♥t♦✉ ♥❛ ♦❜r❛ ❙s✉✲②ü❛♥ ②á✲❝❤✐❡♥ ✭Pr❡❝✐♦s♦ ❡s♣❡❧❤♦ ❞♦s q✉❛tr♦ ❡❧❡♠❡♥t♦s✮ ✉♠❛ té❝♥✐❝❛ ❡s♣❡❝✐❛❧ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉✱ ❜❛s❡❛❞❛ ❡♠ ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s✱ ❞❡ ❣r❛♥❞❡ ♣r❡❝✐sã♦✱ ❞❡♥♦♠✐♥❛❞❛ ♠ét♦❞♦ ❢❛♥✲❢❛♥✱ q✉❡ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❞❡ ❢♦r♠❛ r❡tór✐❝❛ ❡ ❡♥❝♦♥tr❛✈❛ ✉♠❛ ú♥✐❝❛ r❛✐③ ✭♣♦s✐t✐✈❛✮✳
❢❛♥✲❢❛♥✱ r❡❜❛t✐③❛♥❞♦✲♦ ❞❡ ♠ét♦❞♦ ❞❡ ❍♦r♥❡r✳
❖ ♠ét♦❞♦ ❢❛♥✲❢❛♥ ✉s❛❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❤♦❥❡ ❡s❝r✐t❛ ❝♦♠♦x2
+ 252x−5292 = 0✱ ❝♦♥s✐st✐❛ ♥♦ s❡❣✉✐♥t❡✿ ♣❛rt✐❛✲s❡ ❞❡ ✉♠❛ s♦❧✉çã♦ ❛♣r♦①✐♠❛❞❛✱ ♥♦
❝❛s♦✱ x = 19 ✭❛ r❛✐③ ♣♦s✐t✐✈❛ ❞❡ss❛ ❡q✉❛çã♦ ❡stá ❡♥tr❡ 19 ❡ 20✮✱ ❡ ✉s❛✈❛✲s❡ ❛ tr❛♥s❢♦r♠❛çã♦
y = x−19✱ ♣❛r❛ ♦❜t❡r ❛ ❡q✉❛çã♦ y2
+ 290y = 143 ❡♠ y✱ ❝✉❥❛ s♦❧✉çã♦ ❡stá ❡♥tr❡ 0 ❡ 1✳
■❞❡♥t✐✜❝❛♥❞♦y2 ❝♦♠
y✱ ♦❜t✐♥❤❛✲s❡ ✉♠❛ s♦❧✉çã♦ ❛♣r♦①✐♠❛❞❛ ♣❛r❛ ❡ss❛ ❡q✉❛çã♦✿ y = 143 291✱ ❡
❛ss✐♠ ♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ x ❡r❛ ❝♦rr✐❣✐❞♦ ♣❛r❛✿ x = 19 +143
291 = 19,49✳ ❆ ✐❞é✐❛ ❡r❛ r❡♣❡t✐r ♦
♣r♦❝❡ss♦ ❛ ♣❛rt✐r ❞❡ss❡ ♥♦✈♦ r❡s✉❧t❛❞♦ ❛té ❝❤❡❣❛r ❛ ✉♠ ♥ú♠❡r♦ q✉❡ ♥ã♦ ♠❛✐s s❡ ♠♦❞✐✜❝❛ss❡✳ ◆♦ ❝❛s♦✱ ❢❛③❡♥❞♦ z = x−19,49✱ ♦❜t✐♥❤❛✲s❡ ❛ ❡q✉❛çã♦ ❡♠ z✱ z2
+ 290,98z = 0,66 ❡✱ ❞❛í✿
z= 0,66
291,98 = 0,0022✱ ♦ q✉❡ ❥á ❝♦♥✜r♠❛✈❛ ❛s 2❝❛s❛s ❞❡❝✐♠❛✐s ❞♦ ✈❛❧♦r ❡♥❝♦♥tr❛❞♦ ♥♦ ♣❛ss♦
❛♥t❡r✐♦r ✭❝♦♠ ❡❢❡✐t♦✱ ♦s ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❞❡ss❛ r❛✐③ sã♦19,49226✮✳
❊❯❘❖P❆ ❖❈■❉❊◆❚❆▲
❱✐èt❡ ✲ ❈♦❧❡çã♦ ■r♠ã♦s ❇r♦✇♥
❊♠❜♦r❛ ❛✐♥❞❛ ♥ã♦ s❡ ✉s❛ss❡ ♦ ❢♦r♠❛❧✐s♠♦ ❛t✉❛❧✱ ♦ ♣r♦❝❡ss♦ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❛s ❛t✉❛✐s ❡q✉❛çõ❡s ❞♦ ✷➦ ❣r❛✉ r❡s✉♠✐❛✲s❡ ♥❛ r❡❝❡✐t❛ ✉s❛❞❛ ♣♦r ❇❤❛s❦❛r❛✳ ❉♦ sé❝✉❧♦ ❳❱ ❛♦ ❳❱■■✱ ♠✉✐t♦s ❢♦r❛♠ ♦s ♠❛t❡♠át✐❝♦s q✉❡ ❞❡s❡♥✈♦❧✈❡r❛♠ ❢♦r♠❛s ❞✐st✐♥t❛s ❞❡ r❡♣r❡s❡♥t❛r ❛ r❡s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉✳
P❛r❛ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ x2
+ 2ax = b✱ ❋r❛♥ç♦✐s ❱✐èt❡ ✭✶✺✹✵✲✶✻✵✸✮ ♣r♦♣ôs ✉♠❛ ♠✉❞❛♥ç❛
❞❡ ✈❛r✐á✈❡✐s✱ q✉❡ tr❛♥s❢♦r♠❛✈❛ ❛ ❡q✉❛çã♦ ✐♥✐❝✐❛❧ ❡♠ ✉♠❛ ❡q✉❛çã♦ ✐♥❝♦♠♣❧❡t❛✳ ❖s ♣❛ss♦s ♣♦r ❡❧❡ ✉t✐❧✐③❛❞♦s✱ ♥❛ ♥♦t❛çã♦ ❛t✉❛❧✱ sã♦✿
✶✳ ❙❡❥❛ x+a=u
✷✳ ❊♥tã♦ u2
=x2
+ 2ax+a2
✸✳ P❡❧❛ ❡q✉❛çã♦ ❞❛❞❛ x2
+ 2ax =b, ♦✉ s❡❥❛✱ u2
=b+a2
✹✳ ▲♦❣♦ (x+a)2
=u2
=b+a2 ❡
x=√b+a2
−a.
P❛r❛ ✉♠❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞❛ ❢♦r♠❛ ax2
+bx+c= 0,♦ ♠ét♦❞♦ ❞❡ ❱✐èt❡ s❡r✐❛✿
✶✳ ❙❡❥❛ x=u+z
✷✳ ❊♥tã♦ s✉❜st✐t✉✐♥❞♦ ❡♠ ax2
+bx+c= 0, t❡♠✲s❡ a(u+z)2
+b(u+z) +c= 0, ♦✉ s❡❥❛✱ au2
+ (2az+b)u+ (az2
+bz+c) = 0.
✸✳ ❙❡ 2az+b = 0, t❡♠✲s❡ z = −b 2a.
✹✳ ❙✉❜st✐t✉✐♥❞♦ z = −b
2a ❡♠ au
2
+ (2az+b)u+ (az2
+bz+c) = 0, t❡♠✲s❡
au2
+
b2
4a − b2
2a +c
= 0, ♦✉ s❡❥❛✱ au2
= b
2
2a − b2
4a −c= b2
−4ac
4a , ♦✉ ❛✐♥❞❛✱ u =
±
r
b2
−4ac
4a
✺✳ ❋✐♥❛❧♠❡♥t❡ s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s z = −b
2a ❡ u = ±
r
b2
−4ac
4a ❡♠ x =u+z, t❡♠✲s❡
x= −b 2a ±
r
b2
−4ac
4a ,♦✉ s❡❥❛✱ x=
−b±√b2
−4ac
2a .
▼ét♦❞♦ ❞❡ ❉❡s❝❛rt❡s
❉❡s❝❛rt❡s ✲ ▼❛♥s❡❧❧ ❈♦❧❧❡❝t✐♦♥
❊♠ ✶✻✸✼✱ ❘❡♥é ❉❡s❝❛rt❡s ✭✶✺✾✻✲✶✻✺✵✮✱ ❛❧é♠ ❞❡ ♣♦ss✉✐r ✉♠❛ ♥♦✲ t❛çã♦ q✉❡ ❞✐❢❡r✐❛ ❞❛ ❛t✉❛❧ s♦♠❡♥t❡ ♣❡❧♦ sí♠❜♦❧♦ ❞❡ ✐❣✉❛❧❞❛❞❡✱ ❞❡✲ s❡♥✈♦❧✈❡✉ ✉♠ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞❛ r❛✐③ ♣♦s✐t✐✈❛✳ ◆♦ ❛♣ê♥❞✐❝❡ ▲❛ ●é♦♠étr✐❡ ❞❡ s✉❛ ♦❜r❛ ❖ ❉✐s❝✉rs♦ ❞♦ ▼ét♦❞♦✱ ❉❡s❝❛rt❡s r❡s♦❧✈❡✉ ❡q✉❛çõ❡s ❞♦ t✐♣♦✿ x2
=bx+c2
✱ x2
= c2
−bx
❡ x2
= bx−c2
✱ s❡♠♣r❡ ❝♦♠ b ❡ c ♣♦s✐t✐✈♦s✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛
r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞♦ t✐♣♦x2
=bx+c2✱ ✉s♦✉ ♦ s❡❣✉✐♥t❡ ♠ét♦❞♦✿
❚r❛ç❛✲s❡ ✉♠ s❡❣♠❡♥t♦LM✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦c✱ ❡✱ ❡♠L✱ ❧❡✈❛♥t❛✲s❡ ✉♠ s❡❣♠❡♥t♦N L✐❣✉❛❧
❛ b
2❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ LM✳ ❈♦♠ ❝❡♥tr♦ ❡♠ N✱ ❝♦♥stró✐✲s❡ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦
b
2 ❡ tr❛ç❛✲s❡ ❛
r❡t❛ ♣♦rM ❡ N,q✉❡ ❝♦rt❛ ♦ ❝ír❝✉❧♦ ❡♠ O ❡ P.
❖ ◆ ▲ b/2 ❝ ▼ P
t❡♠✲s❡✿
x− b
2
2
=
b
2
2
+c2 ❡ ❞❛í✿
x2
−bx = c2✳ ❍♦❥❡✱ s❛❜❡✲s❡ q✉❡ ❛ s❡❣✉♥❞❛ r❛✐③ é
−OM✱ ♠❛s ❉❡s❝❛rt❡s ♥ã♦ ❝♦♥s✐❞❡r❛✈❛ ❛ r❛✐③ ♥❡❣❛t✐✈❛✳
▼ét♦❞♦ ❞❡ ▲❡s❧✐❡✿
◆♦ sé❝✉❧♦ ❳❱■■■ ♦ ✐♥❣❧ês ❙✐r ❏♦❤♥ ▲❡s❧✐❡✱ ❡♠ s✉❛ ♦❜r❛ ❊❧❡♠❡♥ts ♦❢ ●❡♦♠❡tr②✱ ❛♣r❡s❡♥t♦✉ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦✿ ❉❛❞❛ ✉♠❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛x2
−bx+c= 0✱ s♦❜r❡ ✉♠ s✐st❡♠❛
❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥♦✱ ♠❛r❝❛♠✲s❡ ♦s ♣♦♥t♦s A = (0,1) ❡ B = (b, c)✳ ❚r❛ç❛✲s❡ ♦ ❝ír❝✉❧♦
❞❡ ❞✐â♠❡tr♦AB✳ ❆s ❛❜s❝✐ss❛s ❞♦s ♣♦♥t♦s ❡♠ q✉❡ ❡ss❡ ❝ír❝✉❧♦ ❝♦rt❛r ♦ ❡✐①♦x✱ s❡ ❝♦rt❛r✱ sã♦
❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ q✉❛❞rát✐❝❛ ❞❛❞❛✳
c B = (b, c)
b A= (0,1)
M N
❈♦♠ ❡❢❡✐t♦✱ ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ tr❛ç❛❞❛ é
x− b
2
2
+
y− c+ 1
2
2
=
b
2
2
+
c+ 1
2 −1
2
❡✱ q✉❛♥❞♦y = 0, t❡♠✲s❡x2
−bx =−c.
❈♦♥❝❧✉sã♦
❆t✉❛❧♠❡♥t❡✱ ❛♦ s❡ ❡st✉❞❛r ❛ ❡q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉✱ ✉s❛✲s❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❧✐t❡rár✐❛✱ ❤❡r❞❛❞❛ ❞♦s ❡✉r♦♣❡✉s✱ ❡ ❛ r❡s♦❧✉çã♦ ❢♦r♥❡❝✐❞❛ ♣❡❧♦s ♠ét♦❞♦s ❞♦s ❤✐♥❞✉s ❡ ❞♦s ár❛❜❡s✳
P♦r ♦✉tr♦ ❧❛❞♦✱ ❛❧é♠ ❞❡ ❡st✉❞♦s ❛❧❣é❜r✐❝♦s ❡ ❣❡♦♠étr✐❝♦s✱ ❢♦r❛♠ ❛♣r✐♠♦r❛❞♦s ♠ét♦❞♦s ❞❡ ❝á❧❝✉❧♦s ❛♣r♦①✐♠❛❞♦s✱ ❛ ❡①❡♠♣❧♦ ❞♦ ❛♥t❡♣❛ss❛❞♦ ♣r♦❝❡ss♦ ❞❛ ✏❢❛❧s❛ ♣♦s✐çã♦✑✱ ❡♥❝♦♥tr❛❞♦ ♥♦ ❊❣✐t♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✳ ❚❛✐s ♠ét♦❞♦s t♦r♥❛r❛♠✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❡✜❝❛③❡s ❝♦♠ ♦ ❛❞✈❡♥t♦ ❡ ♣♦♣✉❧❛r✐③❛çã♦ ❞❛s ❝❛❧❝✉❧❛❞♦r❛s ❡ ❞♦s ❝♦♠♣✉t❛❞♦r❡s✳
❘❡❢❡rê♥❝✐❛s
❬❇❖❨❊❘✲✶✾✾✻❪ ❇❖❨❊❘✱ ❈✳❇✳✱ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ✷✳ ❡❞✳ ❙ã♦ P❛✉❧♦✿ ❊❞❣❛r❞ ❇❧✉❝❤❡r✱ ✶✾✾✻✱ ✹✽✽♣✳ ❚r❛❞✉çã♦✿ ❊❧③❛ ❋✳ ●♦♠✐❞❡✳
❬❉❖▼■◆●❯❊❙✲✷✵✵✵❪ ❉❖▼■◆●❯❊❙✱ ❍✳ ❍✳✱ ❙í♥t❡s❡ ❞❛ ❍✐stór✐❛ ❞❛s ❊q✉❛çõ❡s ❆❧❣é❜r✐❝❛s✱ ■♥✳ ❈❛❞❡r♥♦✿ ❊♥s✐♥♦ ❆♣r❡♥❞✐③❛❣❡♠ ❞❡ ▼❛t❡♠át✐❝❛✱ P✉❜❧✐❝❛çõ❡s ❞❛ ❙❇❊▼✲❙P✱ ♥➦✷✱ ✷✵✵✵✳
❬❊❱❊❙✲✶✾✾✺❪ ❊❱❊❙✱ ❍✳✱ ■♥tr♦❞✉çã♦ à ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✱ ❈❛♠♣✐♥❛s✿ ❊❞✐t♦r❛ ❞❛ ❯♥✐❝❛♠♣✱ ✶✾✾✺✱ ✽✹✸♣✳ ❚r❛❞✉çã♦✿ ❍②❣✐♥♦ ❍✳ ❉♦♠✐♥❣✉❡s✳
❬❋❘❆●❖❙❖✲✷✵✵✵❪ ❋❘❆●❖❙❖✱ ❲✳ ❈✳✱ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ ❞❛ ❊q✉❛çã♦ ❞♦ ✷➦ ❣r❛✉✱ ■♥✳ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ♥➦✹✸✱ ✷✵✵✵✱ ♣✳ ✷✵✲✷✺✳
❬▼■❖❘■▼✲✷✵✵✶❪ ▼■❖❘■▼✱ ▼✳ ❆✳✱ ❈❆❘❱❆▲❍❖✱ ❋✳✱ ❇❆❘❖◆❊✱ ❏✳✱ ▼❯◆❙■●◆❆❚❚■ ❏❘✳ ❡ ❇❊●■❆❚❖✱ ❘✳ ●✳✱ P♦r q✉❡ ❇❤❛s❦❛r❛❄✱ ■♥✳ ❍✐stór✐❛ ❡ ❊❞✉❝❛çã♦ ▼❛t❡♠át✐❝❛✱ ✈✳✷✱ ♥➦✷✱ ✷✵✵✶✱ ♣✳✶✷✸✲✶✼✶✳
