Função quadrática

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛ ❡ ❞❛ ❚❡rr❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❏ó❜s♦♥ ❍✉❣♦ ❞❡ ❙♦✉s❛ ❙♦❛r❡s

❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❏ó❜s♦♥ ❍✉❣♦ ❞❡ ❙♦✉s❛ ❙♦❛r❡s

❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ ❡♠ ❝✉♠♣r✐♠❡♥t♦ ❝♦♠ ❛s ❡①✐❣ê♥❝✐❛s ❧❡❣❛✐s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛

❖r✐❡♥t❛❞♦r❛✿

Pr♦❢

_{✳ ❉r}

_{✳ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s}

❏ó❜s♦♥ ❍✉❣♦ ❞❡ ❙♦✉s❛ ❙♦❛r❡s

❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡✱ ❡♠ ❝✉♠♣r✐♠❡♥t♦ ❝♦♠ ❛s ❡①✐❣ê♥❝✐❛s ❧❡❣❛✐s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❊♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛

❆♣r♦✈❛❞♦ ❡♠✿ ✴ ✴

❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿

Pr♦❢a_{✳ ❉r}a_{✳ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s} ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❘◆

❖r✐❡♥t❛❞♦r❛

Pr♦❢✳ ❉r✳ ❋á❣♥❡r ▲❡♠♦s ❙❛♥t❛♥❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❘◆

❊①❛♠✐♥❛❞♦r ■♥t❡r♥♦

Pr♦❢✳ ❉r✳ ❱✐❝t♦r ●✐r❛❧❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❘❏

❆❣r❛❞❡❝✐♠❡♥t♦s

✲ ❆♦s ♠❡✉s ♣❛✐s ♣❡❧♦ ❡s❢♦rç♦ ❡ ❞❡❞✐❝❛çã♦ ♣❛r❛ q✉❡ ❡✉ t✐✈❡ss❡ ❛ ♠❡❧❤♦r ❡❞✉❝❛çã♦ ♣♦ssí✲ ✈❡❧✱ ♣♦r s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❡♠ ❡ ❞❛r ♦ s✉♣♦rt❡ ♥❡❝❡ssár✐♦ ♣❛r❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳ ❚❡♥❤♦ ❝❡rt❡③❛ q✉❡ s❡♠ ❡❧❡s✱ ❡✉ ♥ã♦ t❡r✐❛ s✐❞♦ ❝❛♣❛③ ❞❡ r❡❛❧✐③❛r ❡ss❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

✲ ❆ ♠✐♥❤❛ ❡s♣♦s❛ ♣♦r ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡str❛❞♦✱ ♥❛s ♠❛❞r✉❣❛❞❛s ❞❡ ❡st✉❞♦ ❡ ♠❡ ❞❛♥❞♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝✐❡s✳

✲ ❆♦s ♠❡✉s ❛♠✐❣♦s ❞❛ ♠✐♥❤❛ ❝✐❞❛❞❡ ❙❛♥t❛♥❛ ❞♦ ▼❛t♦s q✉❡ s❡♠♣r❡ ❡st✐✈❡r❡♠ ❝♦♠✐❣♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠ ♠✐♥❤❛ ✐♥❢â♥❝✐❛✳

✲ ❆♦ ♣r♦❢❡ss♦r ❏♦sé ▼❛r✐❛ ●♦♠❡s ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞✐s❝✉ssõ❡s ❡ ♣❡❧♦ ❛❝❡ss♦ à s✉❛ ❜✐❜❧✐♦t❡❝❛ ♣❡ss♦❛❧✳

✲ ❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❣r❛❞✉❛çã♦ ♥❛ ❯❋❘◆✱ q✉❡ ❝♦♥✲ tr✐❜✉✐r❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❞♦❝❡♥t❡✳

✲ ❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❞♦ ■❋❘◆ ✭❝❛♠♣✉s ▼❛❝❛✉ ❡ P❛✉ ❞♦s ❋❡rr♦s✮✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❞✐s❝✉ssõ❡s ❡ ❛♣r❡♥❞✐③❛❞♦✳

✲ ❆♦s ♣r♦❢❡ss♦r❡s ❞❛ Pós✲●r❛❞✉❛çã♦ ♥❛ ❯❋❘◆✿ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐✱ ❉é❜♦r❛✱ ❆♥❞ré ●✉s✲ t❛✈♦✱ ❘♦♥❛❧❞♦✱ ❋❛❣♥❡r ❡ ▼❛r❝❡❧♦ ●♦♠❡s✱ ♣❡❧♦ ❛♠❛❞✉r❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ q✉❡ ❡❧❡s ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦✳

✲ ❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ♣❡❧❛ ❡①❝❡♣❝✐♦♥❛❧ ♦r✐❡♥t❛çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❛❥✉❞❛ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ♣❛r❛ q✉❡ ♣✉❞❡ss❡♠♦s ❝♦♥❝❧✉✐r ❡ss❡ tr❛❜❛❧❤♦✳

✲ ❆♦s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦✱ ❛♣r❡♥❞✐③❛❞♦ ❡ ❞❡s❝♦♥tr❛çã♦✳

✏ ❙❡ ❛s ♣❡ss♦❛s ♥ã♦ ❛❝❤❛♠ ❛ ▼❛t❡♠át✐❝❛ s✐♠♣❧❡s é só ♣♦r q✉❡ ❛✐♥❞❛ ♥ã♦ ♣❡r❝❡❜❡r❛♠ ♦ q✉❛♥t♦ ❛ ✈✐❞❛ é ❝♦♠♣❧✐❝❛❞❛✳✑

❏♦❤♥ ✈♦♥ ◆❡✉♠❛♥♥

❘❡s✉♠♦

❊♠ ❣❡r❛❧✱ ♦ ❡st✉❞♦ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s é ❜❛s❡❛❞♦ ♥✉♠❛ q✉❛♥t✐❞❛❞❡ ❡①❝❡ss✐✈❛ ❞❡ ❢ór♠✉❧❛s✱ t♦❞♦ ❝♦♥t❡ú❞♦ é ❛❜♦r❞❛❞♦ s❡♠ ❥✉st✐✜❝❛t✐✈❛s✳ ❆♣r❡s❡♥t❛♠♦s ❛ ❢✉♥çã♦ q✉❛❞rá✲ t✐❝❛ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❛ ♣❛rt✐r ❞❡ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡ ❞❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦✳ P❛rt✐♥❞♦ ❞❛s ❞❡✜♥✐çõ❡s ♠♦str❛r❡♠♦s q✉❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ❛ ♣❛rá❜♦❧❛ ❡ t❡r♠✐♥❛♠♦s ♥♦ss♦ ❡st✉❞♦s ✈❡r✐✜❝❛♥❞♦ q✉❡ ✈ár✐❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢✉♥çã♦ ♣♦❞❡♠ s❡r ❧✐❞❛s ❛ ♣❛rt✐r ❞❛ s✐♠♣❧❡s ♦❜s❡r✈❛çã♦ ❞♦ s❡✉ ❣rá✜❝♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥str✉í♠♦s t♦❞♦ ♦ ❛ss✉♥t♦ ❥✉st✐✜❝❛♥❞♦ ❝❛❞❛ ♣❛ss♦✱ ❛❜❛♥❞♦♥❛♥❞♦ ♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ❞❡❝♦r❛❞❛s ❡ ♣r❡③❛♥❞♦ ♣❡❧♦ r❛❝✐♦❝í♥✐♦✳

❆❜str❛❝t

■♥ ❣❡♥❡r❛❧✱ t❤❡ st✉❞② ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ✐s ❜❛s❡❞ ♦♥ ❛♥ ❡①❝❡ss✐✈❡ ❛♠♦✉♥t ❢♦r♠✉❧❛s✱ ❛❧❧ ❝♦♥t❡♥t ✐s ❛♣♣r♦❛❝❤❡❞ ✇✐t❤♦✉t ❥✉st✐✜❝❛t✐♦♥✳ ❍❡r❡ ✐s t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ❛♥❞ ✐ts ♣r♦♣❡rt✐❡s ❢r♦♠ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ❛♥❞ t❤❡ t❡❝❤♥✐q✉❡ ♦❢ ❝♦♠♣❧❡t✐♥❣ t❤❡ sq✉❛r❡✳ ❇❛s❡❞ ♦♥ t❤❡ ❞❡✜♥✐t✐♦♥s ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥ ✐s t❤❡ ♣❛r❛❜♦❧❛ ❛♥❞ ✜♥✐s❤❡❞ ♦✉r st✉❞✐❡s ✜♥❞✐♥❣ t❤❛t s❡✈❡r❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❛❞ ❢r♦♠ t❤❡ s✐♠♣❧❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ ②♦✉r ❝❤❛rt✳ ❚❤✉s✱ ✇❡ ❜✉✐❧t t❤❡ ✇❤♦❧❡ ♠❛tt❡r ❥✉st✐❢②✐♥❣ ❡❛❝❤ st❡♣✱ ❛❜❛♥❞♦♥✐♥❣ t❤❡ ✉s❡ ♦❢ ❞❡❝♦r❛t❡❞ ❢♦r♠✉❧❛s ❛♥❞ ✈❛❧✉✐♥❣ t❤❡ r❡❛s♦♥✐♥❣✳

❑❡②✇♦r❞s✿ ◗✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥✳ ❈❛♥♦♥✐❝❛❧ ❢♦r♠✳ ❋❛❝t♦r❡❞ ❢♦r♠✳ P❛r❛❜❧❡✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✷

✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s ✺

✶✳✶ ▼ét♦❞♦ ❞❡ ❈♦♠♣❧❡t❛r ◗✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✶✸

✷✳✶ ❋♦r♠❛ ❈❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ❋♦r♠❛ ❋❛t♦r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸ ●rá❢✐❝♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✷✶

✸✳✶ ❖ ❣rá✜❝♦ ❞❡ f(x) = ax2 _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻}

✸✳✷ ❖ ❣rá✜❝♦ ❞❡ f(x) = ax2+y0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸✳✸ ❖ ❣rá✜❝♦ ❞❡ f(x) = a(x₋x0)2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸✳✹ ❖ ❣rá✜❝♦ ❞❡ f(x) = a(x₋x0)2+y0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✹ ❊st✉❞♦ ❞♦ ●rá✜❝♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛ ✸✷

✹✳✶ ❩❡r♦s ❞❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✷ ❊st✉❞♦ ❞♦ s✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✸ ❊✐①♦ ❞❡ s✐♠❡tr✐❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✸✾

■♥tr♦❞✉çã♦

❊ss❡ tr❛❜❛❧❤♦ é ❞❡st✐♥❛❞♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❡ ♣❛r❛ ❛❧✉♥♦s ❞❡ ❧✐❝❡♥❝✐❛✲ t✉r❛ ❡♠ ♠❛t❡♠át✐❝❛✱ q✉❡ ♣♦ss✉❛♠ ✐♥t❡r❡ss❡ ❡♠ ♠♦❞✐✜❝❛r ♦ tr❛❞✐❝✐♦♥❛❧ ♠ét♦❞♦ ❞❡ ❧❡❝✐♦♥❛r ♦ ❝♦♥t❡ú❞♦ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ ♥♦r♠❛❧♠❡♥t❡ ❡♥s✐♥❛❞♦ ♥♦ ✾♦_{❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ ❡} ♥♦ ✶♦_{❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳}

❆♥❛❧✐s❛♥❞♦ ♦s ❝♦♥t❡ú❞♦s ❝♦♥t✐❞♦s ♥♦s ❝✉rrí❝✉❧♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ♦❝✉♣❛♠ ✉♠ ❧✉❣❛r ❞❡ ❞❡st❛q✉❡✱ ✈✐st♦ q✉❡✱ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❡s❝♦❧❛s✱ é ♦ ❛ss✉♥t♦ ❞❡ ♣r❛t✐❝❛✲ ♠❡♥t❡ t♦❞❛ ✶❛ _{sér✐❡ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ s❡♥❞♦ t❛♠❜é♠ ❜❛s❡ ♣❛r❛ ❛s ❞✐s❝✐♣❧✐♥❛s ❞❡ ❝á❧❝✉❧♦} ❞❛s ✉♥✐✈❡rs✐❞❛❞❡s ❛♣❛r❡❝❡♥❞♦ ♥❛t✉r❛❧♠❡♥t❡ ❡♠ s✐t✉❛çõ❡s ♣rát✐❝❛s ♥♦ ❝♦t✐❞✐❛♥♦✳

❆ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s é ❡♥❛❧t❡❝✐❞❛ ♥♦s ♣❛râ♠❡tr♦s ❝✉rr✐❝✉❧❛r❡s ♥❛❝✐♦✲ ♥❛✐s ❞❡ ♠❛t❡♠át✐❝❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡♠ ❬✷❪

❖ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ♣❡r♠✐t❡ ❛♦ ❛❧✉♥♦ ❛❞q✉✐r✐r ❛ ❧✐♥❣✉❛❣❡♠ ❛❧❣é❜r✐❝❛ ❝♦♠♦ ❛ ❧✐♥❣✉❛❣❡♠ ❞❛s ❝✐ê♥❝✐❛s✱ ♥❡❝❡ssár✐❛ ♣❛r❛ ❡①♣r❡ss❛r ❛ r❡❧❛çã♦ ❡♥tr❡ ❣r❛♥❞❡③❛s ❡ ♠♦❞❡❧❛r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛✱ ❝♦♥str✉✐♥❞♦ ♠♦❞❡❧♦s ❞❡s❝r✐t✐✈♦s ❞❡ ❢❡♥ô♠❡♥♦s ❡ ♣❡r♠✐t✐♥❞♦ ✈ár✐❛s ❝♦♥❡①õ❡s ❞❡♥tr♦ ❡ ❢♦r❛ ❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛✳ ❆ss✐♠✱ ❛ ê♥❢❛s❡ ❞♦ ❡st✉❞♦ ❞❛s ❞✐❢❡r❡♥t❡s ❢✉♥çõ❡s ❞❡✈❡ ❡st❛r ♥♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❡ ❡♠ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡♠ r❡❧❛çã♦ às ♦♣❡r❛çõ❡s✱ ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ s❡✉s ❣rá✜❝♦s ❡ ♥❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛s ❢✉♥çõ❡s✳ ✭♣✳✶✷✶✮✳

❯♠❛ ♣r♦❜❧❡♠át✐❝❛ ❡♥❢r❡♥t❛❞❛ ♥♦ ❝♦t✐❞✐❛♥♦ ❞♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s é ❛ ❢❛❧t❛ ❞❡ ♠❛✲ t❡r✐❛❧✱ ❞❡ q✉❛❧✐❞❛❞❡✱ ♣❛r❛ ❛♣♦✐♦ ♥♦ ❝♦♥t❡ú❞♦ ❛ s❡r ❧❡❝✐♦♥❛❞♦✱ ✉♠❛ ✈❡③ q✉❡✱ ❛ ♠❛✐♦r✐❛ ❞♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡✐①❛♠ ❛ ❞❡s❡❥❛r ❡♠ ✈ár✐♦s ❛s♣❡❝t♦s✳ ❈♦♠♦ ❝✐t❛ ❊❧♦♥ ❡♠ ❬✺❪ ♣✳ ✶✼✵✳

✭✳ ✳ ✳ ✮ ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ✉s❛❞♦s ♥❛s ❡s❝♦❧❛s ❞❡ ✺❛ _{à ✽}❛ _sér✐❡

❛♣r❡s❡♥t❛♠ ❞❡✜❝✐ê♥❝✐❛s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ à ♦❜❥❡t✐✈✐❞❛❞❡✱ às ❛♣❧✐❝❛çõ❡s✱ à ♦❢❡rt❛ ❞❡ ♣r♦❜❧❡♠❛s ❛tr❛❡♥t❡s ❡ ❛♦ ✉s♦ ❞❡ r❛❝✐♦❝í♥✐♦ ❞❡❞✉t✐✈♦✳ ▼❛s✱ ❞❡ ✉♠ ♠♦❞♦ ❣❡r❛❧✱ ♥ã♦ ❛♣r❡s❡♥t❛♠ ❣r❛✈❡s ❡rr♦s ♠❛t❡♠át✐❝♦s✳

■♥❢❡❧✐③♠❡♥t❡ ♥ã♦ s❡ ♣♦❞❡ ❞✐③❡r ♦ ♠❡s♠♦ ❛ r❡s♣❡✐t♦ ❞♦s ❧✐✈r♦s ❞❡st✐♥❛❞♦s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ▼✉✐t♦s ❞❡❧❡s ❛♣r❡s❡♥t❛♠ sér✐♦s ❡rr♦s✳ P♦ss♦ ♠❡s♠♦ ❛✜r♠❛r q✉❡ ♥❡♥❤✉♠ ❞♦s ❧✐✈r♦s q✉❡ ❡①❛♠✐♥❡✐ ✭❡ ❢♦r❛♠ ♠✉✐t♦s✮ ❡st❛✈❛ ✐♥t❡✐r❛♠❡♥t❡ ✐s❡♥t♦ ❞❡ ❛✜r♠❛çõ❡s ❢❛❧s❛s ♦✉ ❛r❣✉♠❡♥t♦s ❞❡❢❡✐t✉♦s♦s✳

P❛r❛ s❡r ❥✉st♦✱ ❞❡✈♦ ❛❞♠✐t✐r q✉❡ ❛❧❣✉♥s ✭❧❛♠❡♥t❛✈❡❧♠❡♥t❡ ♣♦✉❝♦s✮ t❡①t♦s ❛♥❛✲ ❧✐s❛❞♦s ❝♦♥t✐♥❤❛♠ ✉♠ ♥ú♠❡r♦ r❡❞✉③✐❞♦ ❞❡ ❡rr♦s ♠❛t❡♠át✐❝♦s✳

■♥tr♦❞✉çã♦

❆ ♠❛✐♦r✐❛ ♣♦ré♠ tr❛③✐❛ ❞❡✜♥✐çõ❡s✱ r❛❝✐♦❝í♥✐♦s✱ ♠ét♦❞♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡✲ ♠❛s ❡ r❡s♣♦st❛s ✐♥t❡✐r❛♠❡♥t❡ ✐♥❛❞❡q✉❛❞❛s ❡ ❛té ❞❡s♣r♦✈✐❞❛s ❞❡ s✐❣♥✐✜❝❛❞♦✳ ❊ ♦ q✉❡ é ♠❛✐s sér✐♦✿ ♦s ❧✐✈r♦s ❝♦♠ ♠❛✐♦r ♥ú♠❡r♦ ❞❡ ❡rr♦s sã♦ ♦s ♠❛✐s ✈❡♥❞✐❞♦s ♥♦ ♣❛ís✦ ❯♠❛ ♣♦ssí✈❡❧ ❡①♣❧✐❝❛çã♦ ♣❛r❛ ❡st❡ ♣❛r❛❞♦①♦ ❞❡s❛❣r❛❞á✈❡❧ é q✉❡ ❛q✉❡❧❡s ❧✐✈r♦s sã♦ s✐♠♣❧❡s✱ ♥ã♦ ❡①✐❣❡♠ ♠✉✐t♦ r❛❝✐♦❝í♥✐♦✱ ♥ã♦ ❝♦♥té♠ ♣r♦❜❧❡♠❛s ❞✐❢í❝❡✐s ❡ tr❛③❡♠ s♦❧✉çã♦ ❝♦♠♣❧❡t❛ ❞❡ t♦❞❛s ❛s q✉❡stõ❡s ♣r♦♣♦st❛s✱ t♦❞❛s r♦t✐♥❡✐r❛s✳ ❊st❛ ♣♦ssí✈❡❧ r❛③ã♦ ❞♦ s❡✉ ê①✐t♦ ❝♦♠❡r❝✐❛❧ é t❛♠❜é♠ ✉♠ ✐♥❞✐❝❛❞♦r ❞♦ ♥í✈❡❧ ♠é❞✐♦ ❞♦s ♣r♦❢❡ss♦r❡s ❞♦ ♣❛ís✱ q✉❡ ♣r❡❢❡r❡♠ ❡ss❡s t❡①t♦s ♣♦r ♥ã♦ ❧❤❡ ❝❛✉s❛r❡♠ ♦ ❡♠❜❛r❛ç♦ ❞❡ ❝♦♥t❡r❡♠ ♣r♦❜❧❡♠❛s q✉❡ ♥ã♦ s❛❜❡♠ r❡s♦❧✈❡r ♦✉ ❛r❣✉♠❡♥t♦s q✉❡ ♥ã♦ s❛❜❡♠ ❡①♣❧✐❝❛r✳

◆❡st❡ tr❛❜❛❧❤♦ ❜✉s❝❛♠♦s ❡st✉❞❛r ❢✉♥çõ❡s q✉❛❞rát✐❝❛s ❞❡ ✉♠❛ ❢♦r♠❛ ❛♠♣❧❛✱ ❝♦♠❡ç❛♥❞♦ ❝♦♠ s✉❛s ♦r✐❣❡♥s ♥❛ ❝✐✈✐❧✐③❛çã♦ ❜❛❜✐❧ô♥✐❝❛✳ ❊ss❛ ❛♥t✐❣❛ ❝✐✈✐❧✐③❛çã♦ r❡s♦❧✈✐❛ ♣r♦❜❧❡♠❛s✱ q✉❡ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❤♦❥❡ ❡♠ ❞✐❛ ♣♦r ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ♣♦r ✈♦❧t❛ ❞❡ ✶✼✵✵ ❛♥♦s ❛✳❈✳ ❝♦♠ ❛ ♠❡s♠❛ ❢ór♠✉❧❛✱ ♦✉ ♥♦ ❝❛s♦ ❞❡❧❡s r❡❣r❛✱ q✉❡ é ✉t✐❧✐③❛❞❛ ❛té ❤♦❥❡ ❡♠ ♥♦ss❛ ❡s❝♦❧❛ ❜ás✐❝❛✳

❊♠ ❣❡r❛❧✱ ♦ ❝♦♥t❡ú❞♦ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s é ❛♣r❡s❡♥t❛❞♦ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❡❝â♥✐❝❛✱ ❝❛rr❡❣❛❞♦ ❞❡ ❢ór♠✉❧❛s q✉❡ ♦s ❛❧✉♥♦s ♥ã♦ s❛❜❡♠ ❞❡ ♦♥❞❡ ✈❡♠ ❡✱ ❛❧❣✉♠❛s ✈❡③❡s✱ ♦ ♣r♦❢❡ss♦r t❛♠❜é♠ ♥ã♦✳

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ♣❛r❛ ♠♦t✐✈❛r ♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ r❡s❣❛t❛♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❛♥t✐❣♦ ❡ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♦✉tr♦ ♣rát✐❝♦ q✉❡ r❡❝❛❡♠ ❡♠ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉✳ P❛r❛ ❥✉st✐✜❝❛r ❛❧❣✉♥s r❡s✉❧t❛❞♦s✱ ✉t✐❧✐③❛♠♦s ❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦✱ q✉❡ ❝♦st✉♠❛ ♥ã♦ s❡r ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s ❞❛ ❡s❝♦❧❛ ❜ás✐❝❛✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐♠♦s ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ❡st✉❞❛♠♦s ❛ ❋♦r♠❛ ❈❛♥ô♥✐❝❛✳ ❆ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ é ❛ r❡♣r❡s❡♥t❛çã♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❞❡✈✐❞♦ ❛♦ ♥ú♠❡r♦ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ ❡①tr❛í♠♦s ❛ ♣❛rt✐r ❞❡❧❛✳ ■♥❢❡❧✐③♠❡♥t❡ s❡✉ ❡st✉❞♦ é ♣♦✉❝♦ ❡①♣❧♦r❛❞♦ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ♦s ❧✐✈r♦s t❛♥t♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥t♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♣❡❧♦ ♠❡♥♦s ❛ ♠❛✐♦r✐❛ ❞❡❧❡s✱ ♥ã♦ ❢❛③❡♠ ❡ss❛ ❛❜♦r❞❛❣❡♠ ❡ ♦s ♣r♦❢❡ss♦r❡s ❛♦ r❡s♦❧✈❡r❡♠ s❡❣✉✐✲❧♦s✱ t❛♠❜é♠ ♥ã♦✳

P❛r❛ ❡s❝r❡✈❡r ❛ ❢✉♥çã♦ ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ é ♥❡❝❡ssár✐♦ r❡❝♦rr❡r ❛♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ q✉❛❞r❛❞♦s✱ q✉❡ ♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ♦❜t❡r ✉♠ tr✐♥ô♠✐♦ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦ ❡ ❢❛t♦rá✲❧♦✳ ❉❡ss❡ ♠♦❞♦✱ t❛♥t♦ ♥♦ ✾♦ _{❛♥♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ❝♦♠♦ ♥♦ ✶}♦ _{❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ é} ♣♦ssí✈❡❧ ❛❜♦r❞❛r ♦ ❛ss✉♥t♦✳

❆ ♣❛rt✐r ❞❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❞❡t❡r♠✐♥❛♠♦s✿ ♦s ③❡r♦s ❞❛ ❢✉♥çã♦✱ ✈❛❧♦r ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ ❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ❝♦❡✜❝✐❡♥t❡a ♥❛ ❢✉♥çã♦✳

❖✉tr❛ ❢♦r♠❛ ❞❡ s❡ r❡♣r❡s❡♥t❛r ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❡st✉❞❛❞❛ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ é ❛ ❢♦r♠❛ ❢❛t♦r❛❞❛✱ q✉❡ ♥♦s ❞á ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦✱ ❛❧é♠ ❞❡ ♣♦ss✐❜✐❧✐t❛r✱

■♥tr♦❞✉çã♦

❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❛❧❣é❜r✐❝❛✱ ♦ ❡st✉❞♦ ❞♦s s✐♥❛✐s q✉❡ ❛ ❢✉♥çã♦ ♣♦❞❡ ❛ss✉♠✐r✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❡st✉❞❛♠♦s ♦ ❣rá❢✐❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❞❡✜♥✐♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦ q✉❡ é ✉♠❛ ♣❛rá❜♦❧❛ ❡ ❡♠ s❡❣✉✐❞❛ ♠♦str❛♠♦s q✉❡ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ✉♠❛ ♣❛rá❜♦❧❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s ♦ ❡st✉❞♦ ❞❛s tr❛♥s❧❛çõ❡s ❞♦ ❣rá✜❝♦ ❡ ❛♣r❡♥❞❡♠♦s ❝♦♠♦✱ ❛ ♣❛rt✐r ❞❡❧❡✱ ❡st✉❞❛r ♦s ③❡r♦s ❡ ♦ s✐♥❛❧ ❞❛ ❢✉♥çã♦✱ ❡st✉❞♦s ❡ss❡s✱ ❢❡✐t♦s ❞❡ ♠❛♥❡✐r❛ ❛❧❣é❜r✐❝❛ ♥❛s s❡çõ❡s ✷✳✶ ❡ ✷✳✷✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥❝❡rr❛♠♦s ♦ ♥♦ss♦ ❡st✉❞♦ ❝♦♠ ❛ s❡çã♦ ✹✳✸ ♦♥❞❡ ♠♦str❛♠♦s q✉❡ ❛ ♣❛rá❜♦❧❛ ♣♦ss✉✐ ✉♠ ❡✐①♦ ❞❡ s✐♠❡tr✐❛✳ ❖ ✉s♦ ❞❡ s♦❢t✇❛r❡s ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ é ✉t✐❧✐③❛❞♦ ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❞❡ ❛✉①í❧✐♦✱ ♥❛ ❝♦♥str✉çã♦ ❞❡ ❣rá✜❝♦s✳ ❙✉❣❡r✐♠♦s ♦ ✉s♦ ❞♦ s♦❢t✇❛r❡ ❣❡♦❣❡❜r❛✱ ❞✐str✐❜✉í❞♦ ❣r❛t✉✐t❛♠❡♥t❡ ♥❛ ✐♥t❡r♥❡t ❛tr❛✈és ❞♦ ❡♥❞❡r❡ç♦ ❡❧❡trô♥✐❝♦ ✭✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣✮✳

❈❛♣ít✉❧♦ ✶

Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s

◗✉❛❞rát✐❝❛s

❈♦♠❡ç❛r❡♠♦s ♥♦ss♦ ❡st✉❞♦ r❡s♦❧✈❡♥❞♦ ✉♠ ♣r♦❜❧❡♠❛ ❤✐stór✐❝♦ ❡ ♦✉tr♦ ♣rát✐❝♦ ❡♥✈♦❧✲ ✈❡♥❞♦ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❛ss✉♥t♦ ❥á ❝♦♥❤❡❝✐❞♦ ♣❡❧♦s ❛❧✉♥♦s ❡ q✉❡ s❡r✈✐rá ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ❞❡✜♥✐r♠♦s ❢✉♥çã♦ q✉❛❞rát✐❝❛✳

Pr♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡stã♦ ❡♥tr❡ ♦s ♠❛✐s ❛♥t✐❣♦s✿ ❊♥❝♦♥✲ tr❛r ❞♦✐s ♥ú♠❡r♦s a ❡ b✱ ❝♦♥❤❡❝❡♥❞♦ s✉❛ s♦♠❛ ❡ s❡✉ ♣r♦❞✉t♦✱ a+b = s ❡ a.b = p✱ q✉❡

♣♦❞❡♠♦s r❡s♦❧✈❡r ❤♦❥❡ ❡♠ ❞✐❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ sã♦ ❡♥❝♦♥tr❛❞❛s ❡♠ t❛❜❧❡t❡s ❞❛ ❇❛❜✐❧ô♥✐❛✶ _{❛♥t✐❣❛ ❛ q✉❛s❡ ✹✵✵✵ ❛♥♦s✳ ▲❡r ❆s❣❡r ❆❛❜♦❡ ❡♠ ❬✶❪ ♣✳ ✷✹ ❡ ♣✳} ✸✵✳

Pr♦❜❧❡♠❛ ✶✳✶ ✭Pr♦❜❧❡♠❛ ❤✐stór✐❝♦✮✳ ❊♥❝♦♥tr❛r ❞♦✐s ♥ú♠❡r♦s ❝♦♥❤❡❝❡♥❞♦ ❛♣❡♥❛s ❛ s✉❛ s♦♠❛ ❡ ♦ s❡✉ ♣r♦❞✉t♦✳

❙♦❧✉çã♦

❈♦♥s✐❞❡r❡ ❛ s♦♠❛ ❝♦♠♦ s ❡ ♦ ♣r♦❞✉t♦ ❝♦♠♦ p✳ ❙❡♥❞♦ ✉♠ ❞♦s ♥ú♠❡r♦s ♣r♦❝✉r❛❞♦s x

♦ ♦✉tr♦ s❡rás₋x ❡ ❞❡ss❛ ❢♦r♠❛ ♦ ♣r♦❞✉t♦ s❡rá

p= (s₋x)x,

♦✉ s❡❥❛✱

✶_{◗✉❛♥❞♦ ♠❡♥❝✐♦♥❛r♠♦s ❜❛❜✐❧ô♥✐♦s ♦✉ ❝✐✈✐❧✐③❛çã♦ ❜❛❜✐❧ô♥✐❝❛ ❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ ❛♦s ♣♦✈♦s q✉❡ ❤❛✲}

❜✐t❛r❛♠ ❛ r❡❣✐ã♦ ❞❛ ♠❡s♦♣ôt❛♠✐❛✱ ❝♦♠♦ ♦s s✉♠ér✐♦s✱ ♦s ❛❝❛❞✐❛♥♦s✱ ♦s ❝❛❧❞❡✉s✱ ♦s ❛ssír✐♦s ❡♥tr❡ ♦✉tr♦s ♣♦✈♦s ❛♥t✐❣♦s✳ ❯s❛r❡♠♦s ♦ t❡r♠♦ ❜❛❜✐❧ô♥✐♦s✱ ❝♦♠♦ é ❝♦♠✉♠✱ ✭❛ss✐♠ ❝♦♠♦ ❊✈❡s ❡♠ ❬✹❪✮ ♣♦r s✐♠♣❧❡s ❝♦♥✈❡♥✐ê♥❝✐❛✳

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

x2₋sx+p= 0. ✭✶✳✶✮

❯♠ ♦✉tr♦ ❡♥✉♥❝✐❛❞♦ ♣❛r❛ ❡ss❡ ♠❡s♠♦ ♣r♦❜❧❡♠❛ s❡r✐❛ ❞❡t❡r♠✐♥❛r ♦s ❧❛❞♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❝✉❥♦ ♦ s❡♠✐♣❡rí♠❡tr♦ és ❡ ❛ ár❡❛ é a✳ ❈♦♠♦ s✉❣❡r❡ P❛✉❧♦ ❈♦♥t❛❞♦r ❡♠ ❬✸❪✳

❱❛❧❡ s❛❧✐❡♥t❛r q✉❡ r❡♣r❡s❡♥t❛r ❛ s♦❧✉çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ é ❝♦♠♦ ❢❛③❡♠♦s ♥♦s ❞✐❛s ❛t✉❛✐s✱ ♣♦✐s ♦s ♣♦✈♦s ❛♥t✐❣♦s ♥ã♦ ❝♦♥❤❡❝✐❛♠ ❡ss❡ ♠❡❝❛♥✐s♠♦ ❞❡ r❡♣r❡s❡♥t❛çã♦✱ q✉❡ só ❝♦♠❡ç♦✉ ❛ s❡r ✉t✐❧✐③❛❞♦ ♣♦r ❱✐èt❡✷_{✱ ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ✶✻✳ P❛r❛} r❡s♦❧✈❡r ❡ss❡ t✐♣♦ ❞❡ ❡①♣r❡ssã♦ ♦s ❇❛❜✐❧ô♥✐♦s ♣♦ss✉✐❛♠ ✉♠❛ r❡❝❡✐t❛✱ q✉❡ s❡❣✉♥❞♦ ❊❧♦♥ ❡♠ ❬✺❪ ❡r❛ ❡♥✉♥❝✐❛❞❛ ❛ss✐♠✿

❊❧❡✈❡ ❛♦ q✉❛❞r❛❞♦ ❛ ♠❡t❛❞❡ ❞❛ s♦♠❛✱ s✉❜tr❛✐❛ ♦ ♣r♦❞✉t♦ ❡ ❡①tr❛✐❛ ❛ r❛í③ q✉❛❞r❛❞❛ ❞❛ ❞✐❢❡r❡♥ç❛✳ ❙♦♠❡ ❛♦ r❡s✉❧t❛❞♦ ❛ ♠❡t❛❞❡ ❞❛ s♦♠❛✳ ■ss♦ ❞❛rá ♦ ♠❛✐♦r ❞♦s ♥ú♠❡r♦s ♣r♦❝✉r❛❞♦s✳ ❙✉❜tr❛✐❛✲♦ ❞❛ s♦♠❛ ♣❛r❛ ♦❜t❡r ♦ ♦✉tr♦ ♥ú♠❡r♦✳

❊ss❛ r❡❣r❛ ♥♦s ❢♦r♥❡❝❡✱ ❡♠ ♥♦ss❛ ♥♦t❛çã♦ ❛t✉❛❧✱ ♦s ♥ú♠❡r♦s

x= s 2+

r s

2

2

−p ❡ s₋x= s 2−

r s

2

2

−p ✭✶✳✷✮

❖s ♣❛ss♦s ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ♣r♦❝✉r❛❞♦s ♥ã♦ ❡r❛♠ ❥✉st✐✜❝❛❞♦s✱ ♣♦✐s ❡❧❡s ♥ã♦ s❡ ♣r❡♦❝✉♣❛✈❛♠ ❝♦♠ ❞❡♠♦♥str❛çõ❡s✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ❛✉t♦r❡s ❞❡ ❬✻❪ ❤á ✐♥❞í❝✐♦s ❞❡ q✉❡ ♦s ❜❛❜✐❧ô♥✐♦s ❝❤❡❣❛r❛♠ ❛ ❡ss❛s ❡①♣r❡ssõ❡s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❈♦♥s✐❞❡r❛♥❞♦ α ❡ β ♦s ♥ú♠❡r♦s ♣r♦❝✉r❛❞♦s✱ sã♦ ❝♦♥❤❡❝✐❞♦s ❞♦✐s ♥ú♠❡r♦s s ❡ p✱ t❛✐s

q✉❡s=α+β ❡p=αβ✳ ❆ss✐♠✱ ❛♣❡s❛r ❞❡α❡β s❡r❡♠ ❞❡s❝♦♥❤❡❝✐❞♦s✱ ❛ ♠é❞✐❛ ❛r✐t♠ét✐❝❛

α+β

2 =

s

2 é ❝♦♥❤❡❝✐❞❛ ❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r ❡q✉✐❞✐st❛♥t❡ ❞❡ α ❡ ❞❡ β✳

❆❞♠✐t✐♥❞♦ q✉❡ α _≤ β✱ t❡♠♦s q✉❡ s

2 −α = β−

s

2✳ ❈❤❛♠❛♥❞♦ ❡st❛ ❞✐❢❡r❡♥ç❛ ❝♦♠✉♠

❞❡d✱ ♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧ ❞❡ ❡♥❝♦♥tr❛r ♦s ❞♦✐s ♥ú♠❡r♦sα ❡β✱ s❡ r❡❞✉③ ❛ ❡♥❝♦♥tr❛r ♦ ú♥✐❝♦

♥ú♠❡r♦d✱ ♣♦✐sα= s

2 −d ❡β =

s

2 +d✳

✷_{❉❡✈❡✲s❡ ❛♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❋r❛♥ç♦✐s ❱✐èt❡✱ ♥❛s❝✐❞♦ ❡♠ ✶✺✹✵✱ ❛ ✐♥tr♦❞✉çã♦ ❞❛ ♣rát✐❝❛ ❞❡ s❡ ✉s❛r}

✈♦❣❛✐s ♣❛r❛ r❡♣r❡s❡♥t❛r ✐♥❝ó❣♥✐t❛s ❡ ❝♦♥s♦❛♥t❡s ♣❛r❛ r❡♣r❡s❡♥t❛r ❝♦♥st❛♥t❡s ❝♦♠♦ ♠❡♥❝✐♦♥❛ ❊✈❡s ❡♠ ❬✹❪ ♣✳✸✵✾✳

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

❙❛❜❡♠♦s q✉❡✿

p = αβ

= s 2−d

s

2+d

= s 2

2

−d2.

▲♦❣♦

d2 =s 2

2

−p,

❝♦♠♦d é ♥ã♦ ♥❡❣❛t✐✈♦✱

d= r s 2 2 −p. ❉❛í

α= s

2−d=

s 2− r s 2 2 −p ❡

β = s

2 +d =

s 2+ r s 2 2 −p. ❖✉tr❛ ♠❛♥❡✐r❛ ❞❡ ♦❜t❡r ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ♣❡❧♦s ❜❛❜✐❧ô♥✐♦s é ♦ ♠ét♦❞♦ ❞❡ ❝♦♠♣❧❡✲ t❛r q✉❛❞r❛❞♦✱ ❛♦ q✉❛❧ ❞❛r❡♠♦s ê♥❢❛s❡ ♥♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ t❡♥❞♦ ❡♠ ✈✐st❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✳

❆♣❡s❛r ❞❡ t♦❞♦ s❡✉ ✈❛❧♦r✱ ❡ss❡ t❡♠❛ ❛✐♥❞❛ ❡♥❝♦♥tr❛ ❛❧❣✉♠❛ r❡s✐stê♥❝✐❛ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ♣❡❧♦ ♠❡♥♦s ♥❛s sér✐❡s ✐♥✐❝✐❛✐s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❙❡♥❞♦ ❛♣r❡s❡♥t❛❞♦ ❛♣❡♥❛s q✉❛♥❞♦ s❡ tr❛❜❛❧❤❛ ❝♦♠ ❡q✉❛çõ❡s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❡♠ ❣❡r❛❧✱ ♥♦ ú❧t✐♠♦ ❛♥♦✳

◗✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s ♥♦ ✾♦_{❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♣r♦❝✉r❛✲} ♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦s ✈❛❧♦r❡s ❞❡x q✉❡ ❛ r❡s♦❧✈❛♠✳ ❈♦♠❡ç❛♠♦s s❡♠♣r❡ ❝♦♠ ♦s ❝❛s♦s ♠❛✐s

s✐♠♣❧❡s✿

ax2+c= 0,

q✉❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ✐s♦❧❛♥❞♦ ♦ ✈❛❧♦r ❞❡x✿

x=_±

r

−_ac.

❖s ❛❧✉♥♦s tr❛❜❛❧❤❛♠ ❝♦♠ ❡①❡♠♣❧♦s ❝♦♠♦ ❡ss❡✱ ♣♦ré♠ ♥♦ ❝❛s♦ ❞♦ tr✐♥ô♠✐♦ ❡st❛r ❡♠

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

s✉❛ ❢♦r♠❛ ❝♦♠♣❧❡t❛ ♦s ♣r♦❢❡ss♦r❡s ❛❜❛♥❞♦♥❛♠ ❡ss❡ ♠ét♦❞♦ ❞❡ r❡s♦❧✉çã♦ ❡ ❥á ❛♣r❡s❡♥t❛♠ ❛s ❢ór♠✉❧❛s r❡s♦❧✉t✐✈❛s ♣❛r❛ r❡s♦❧✈❡r ❛s ❡q✉❛çõ❡s✳

➱ ✐♥t❡r❡ss❛♥t❡ q✉❡ ♦ ♣r♦❢❡ss♦r tr❛❜❛❧❤❡ ❝❛s♦s ❞♦ t✐♣♦✿

a(x+m)2+k = 0,

♦♥❞❡ ♣♦❞❡♠♦s ✉s❛r ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❛♥t❡r✐♦r ❡ ✐s♦❧❛♥❞♦ ♦ ✈❛❧♦r ❞❡x✱ ❡♥❝♦♥tr❛♠♦s✿

x=₋m_± r

−k_a,

♦✉ s❡❥❛✱ ❛ ✐❞é✐❛ é ♣❛rt✐r ❞❡ ✉♠ tr✐♥ô♠✐♦ ❝♦♠♣❧❡t♦(ax2+bx+c)❡ ❝❤❡❣❛r ❛ ✉♠❛ ❡①♣r❡ssã♦

❝♦♥❤❡❝✐❞❛✱a(x+m)2_{+k✱ q✉❡ s❛❜❡♠♦s ❝♦♠♦ ❞❡t❡r♠✐♥❛r s✉❛ s♦❧✉çã♦✳}

◆❛ ♣ró①✐♠❛ s❡çã♦ ❡st✉❞❛r❡♠♦s ❝♦♠♦ ❝❤❡❣❛r ❛ ❡ss❛ ❡①♣r❡ssã♦ ❝♦♥❤❡❝✐❞❛ ♣❛rt✐♥❞♦ ❞❡ ✉♠ tr✐♥ô♠✐♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❝♦♠♣❧❡t♦✳

✶✳✶ ▼ét♦❞♦ ❞❡ ❈♦♠♣❧❡t❛r ◗✉❛❞r❛❞♦

❚❡♠♦s ♦ tr✐♥ô♠✐♦x2₋sx+p ❡ ♥♦ss♦ ✐♥t✉✐t♦ é ❡s❝r❡✈ê✲❧♦ ❝♦♠♦ ✉♠ q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦

❞♦ t✐♣♦ x2

−2kx+k2_{✳ ❏á ♣♦ss✉í♠♦s✿}

x2₋2s 2x+p,

♠❛s ❢❛❧t❛ ❛ ♣❛r❝❡❧❛ (s

2)

2_{✱ ❡♥tã♦✱ ♣❛r❛ ♥ã♦ ❛❧t❡r❛r♠♦s ♦ tr✐♥ô♠✐♦✱ s♦♠❛♠♦s ❡ s✉❜tr❛í♠♦s}

❡ss❛ ♣❛r❝❡❧❛✿

x2₋2s 2x+

s

2

2

−s

2

2

+p.

❉❡ss❛ ❢♦r♠❛✿

x2₋sx+p=x₋ s

2

2

−s₂

2

+p.

❈♦♠ ❡ss❡ tr✐♥ô♠✐♦ ❢❛t♦r❛❞♦✱ ♣♦❞❡♠♦s ♦❜t❡r ❢❛❝✐❧♠❡♥t❡ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✳ ❉❡ ❢❛t♦✿

x₋ s

2

2

−₂s

2

+p= 0,

é ❡q✉✐✈❛❧❡♥t❡ ❛✿

x₋ s

2 =±

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

♦✉ s❡❥❛✱

x= s 2 ±

r s

2

2

−p.

◗✉❡ é ❛ s♦❧✉çã♦ tr❛❜❛❧❤❛❞❛ ♣❡❧♦s ❜❛❜✐❧ô♥✐♦s ❛ q✉❛s❡ ✹✵✵✵ ❛♥♦s✳

❊①❡♠♣❧♦ ✶✳✶✳

◗✉❛✐s sã♦ ♦s ♥ú♠❡r♦s ❝✉❥❛ ❛ s♦♠❛ é 8 ❡ ♦ ♣r♦❞✉t♦ é 15❄

❙♦❧✉çã♦

◆❡st❡ ❡①❡♠♣❧♦ t❡♠♦ss= 8 ❡p= 15✱ ❛ss✐♠✱ s❡ ✉♠ ❞♦s ♥ú♠❡r♦s éx♦ ♦✉tr♦ s❡rá8₋x✱

❡ ♦ ♣r♦❞✉t♦ éx(8₋x) = 15✱ q✉❡ ❞á ♦r✐❣❡♠ ❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✿

x2₋8x+ 15 = 0.

❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦✱ ♦❜t❡♠♦s✿

(x₋4)2

−16 + 15 = 0,

q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛✿

(x₋4)2 = 1.

❆ss✐♠

x₋4 =_±1,

♣♦rt❛♥t♦✿

x= 5 ♦✉x= 3.

❖❜s❡r✈❡ q✉❡ ❛❝❤❛r ❞♦✐s ♥ú♠❡r♦s ❝✉❥❛ ❛ s♦♠❛ é 8 ❡ ❝✉❥♦ ♦ ♣r♦❞✉t♦ é 15✱ ♥ã♦ é ✉♠❛

t❛r❡❢❛ ❝♦♠♣❧✐❝❛❞❛✱ ❡ s❡♠ ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s✱ ♣♦❞❡rí❛♠♦s ❝❤❡❣❛r ❛♦s ♥ú♠❡r♦s 3❡ 5 ♣♦r

✐♥s♣❡çã♦✳

➱ ✐♠♣♦rt❛♥t❡ q✉❡ ♦ ♣r♦❢❡ss♦r ❡①♣❧♦r❡ ❡ss❛ ♣rát✐❝❛ ❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♣♦r ✐♥s♣❡çã♦ ♦✉ t❡♥t❛t✐✈❛✱ ❞❛♥❞♦ ❧✐❜❡r❞❛❞❡ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ♣r♦❝✉r❡ s♦❧✉çõ❡s s❡♠ ✉t✐❧✐③❛r s❡✉ ❝❛❞❡r♥♦✱ ❡st✐♠✉❧❛♥❞♦ ❛ss✐♠ ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ❡ ♦ ❝á❧❝✉❧♦ ♠❡♥t❛❧ q✉❡ é ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛ ❞♦ ❛❧✉♥♦✳ ➱ ♥❡❝❡ssár✐♦ q✉❡ ♦ ♣r♦❢❡ss♦r ❛❝♦♠♣❛♥❤❡ ❛t❡♥t❛♠❡♥t❡ ❡ss❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ♣♦✐s ♥❡♠ s❡♠♣r❡ é tã♦ s✐♠♣❧❡s r❡s♦❧✈❡r ❡q✉❛çõ❡s ♣♦r ♠❡✐♦ ❞❡ss❡ ♠ét♦❞♦✳

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

P♦r ❡①❡♠♣❧♦✱ s❡ q✉❡r❡♠♦s ❞♦✐s ♥ú♠❡r♦s✱ ❝✉❥❛ s♦♠❛ é1❡ ❝✉❥♦ ♣r♦❞✉t♦ é₋1✱ ❛ r❡s♣♦st❛

♥ã♦ é tã♦ ❡❧❡♠❡♥t❛r ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ❡ss❡s ♥ú♠❡r♦s ♥ã♦ sã♦ ✐♥t❡✐r♦s✱ ♦ q✉❡ ❛❝❛❜❛ ❞✐✜❝✉❧t❛♥❞♦ ♦ ❝á❧❝✉❧♦ ♠❡♥t❛❧✳

P❛r❛ s= 1 ❡ p=₋1✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

x(1₋x) = ₋1,

♦✉ s❡❥❛✱

x2₋x₋1 = 0.

❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦✱ t❡♠♦s

x₋ 1

2

2

− 1₄ −1 = 0,

q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛

x₋ 1

2

2

− 5₄ = 0.

❆ss✐♠

x₋1

2 =±

√

5 2 ,

❧♦❣♦

x= 1 +

√

5

2 ♦✉x=

1₋√5 2 .

❖ ✈❛❧♦r ♣♦s✐t✐✈♦ ❡♥❝♦♥tr❛❞♦ ♣❛r❛ x é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♥ú♠❡r♦ ❞❡ ♦✉r♦✳✸

❖ ♣r♦❜❧❡♠❛ ❛ s❡❣✉✐r é ✉♠❛ s✐t✉❛çã♦ ♣rát✐❝❛✱ q✉❡ ✐♥✐❝✐❛❧♠❡♥t❡ ♣❛r❡❝❡ s❡r s♦♠❡♥t❡ ❞❡ ❣❡♦♠❡tr✐❛✱ ♠❛s q✉❡ r❡❝❛✐ ❡♠ ✉♠❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❛♦ ❜✉s❝❛r♠♦s ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ár❡❛✳

Pr♦❜❧❡♠❛ ✶✳✷✳ ▼❡✉ ❛✈ô ♣♦ss✉✐ ✉♠ t❡rr❡♥♦ r❡t❛♥❣✉❧❛r ❝♦♠ ❞✐♠❡♥sõ❡s 22m ♣♦r 30m✳

❙❛❜❡✲s❡ q✉❡ ✉♠ ❞♦s ❧❛❞♦s q✉❡ ♠❡❞❡22m ❥á ♣♦ss✉✐ ✉♠ ♠✉r♦ ❝♦♥str✉í❞♦ ❡ ❡❧❡ q✉❡r ✉t✐❧✐③❛r

♣❛rt❡ ❞❡ss❡ ♠✉r♦ ♣❛r❛ ❢❛③❡r ✉♠ ❝❡r❝❛❞♦ r❡t❛♥❣✉❧❛r ❞❡ 48m2_{✳ ❉✐s♣♦♥❞♦ ❞❡}

28m ❞❡ t❡❧❛ é

♣♦ssí✈❡❧ ❝♦♥str✉✐r ❡ss❡ ❝❡r❝❛❞♦❄ ◗✉❛✐s sã♦ ❛s ♠❡❞✐❞❛s ❞♦s s❡✉s ❧❛❞♦s❄

✸_{❖ ♥ú♠❡r♦ ❞❡ ♦✉r♦ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ sí♠❜♦❧♦ ❞❡ ❤❛r♠♦♥✐❛✳ ❆♣❛r❡❝❡ ♥❛ ♥❛t✉r❡③❛✱ ♥❛ ❛rt❡✱ ❛rq✉✐t❡t✉r❛✱}

♠ús✐❝❛ ❡ ♥♦s s❡r❡s ❤✉♠❛♥♦s✳ P♦r ❡①❡♠♣❧♦ ❛ r❛③ã♦ ❡♥tr❡ ✉♠ t❡r♠♦ ❡ s❡✉ ❛♥t❡❝❡ss♦r ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ❋✐❜♦♥❛❝❝✐ ✭✶✱ ✶✱ ✷✱ ✸✱ ✺✱ ✽✱ ✶✸✱ ✷✶✱ ✳✳✳✮ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ♥ú♠❡r♦ ❞❡ ♦✉r♦✳

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

❙♦❧✉çã♦

❙❡ ❝❤❛♠❛r♠♦s ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ ❞♦s ❧❛❞♦s ❞♦ ❝❡r❝❛❞♦ ❞❡x✱ ❛ ♦✉tr❛ ♠❡❞✐❞❛ s❡rá(28₋2x)✳

❆❞♠✐t✐♥❞♦ q✉❡ ♦ ❧❛❞♦ q✉❡ ♠❡❞❡xé ♦ ♥ã♦ ♣❛r❛❧❡❧♦ ❛♦ ♠✉r♦✱ ❛ ár❡❛ ❞♦ ❝❡r❝❛❞♦ éx(28₋2x)

❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

x(28₋2x) = 48, ♦✉ s❡❥❛✱ x2₋14x+ 24 = 0.

❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦✿

(x₋7)2

−49 + 24 = 0, ♦ q✉❡ ♥♦s ❧❡✈❛ ❛(x₋7)2

= 25,

❡ ❛ss✐♠

x₋7 =_±5,

♦✉ s❡❥❛✱

x= 12 ♦✉ x= 2.

❊♥❝♦♥tr❛♠♦s ❞✉❛s s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦ x2

−14x+ 24 = 0✱ ♣♦ré♠ só ✉♠❛ ❞❡❧❛s

s❛t✐s❢❛③ ♥♦ss♦ ♣r♦❜❧❡♠❛✱ ♣♦✐s ♣❛r❛x= 2✱ ♦ ❝❡r❝❛❞♦ t❡rá ❞✐♠❡♥sõ❡s 24m ① 2m✱ ♦ q✉❡ ♥ã♦

é ♣♦ssí✈❡❧✱ ❥á q✉❡ ♦ ♠✉r♦ só ♣♦ss✉✐ 22m ❞❡ ❡①t❡♥sã♦✳

❉❡ss❡ ♠♦❞♦✱ ❛ ú♥✐❝❛ s♦❧✉çã♦ ♣♦ssí✈❡❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ é x = 12✱ t❡r❡♠♦s ❛ss✐♠ ✉♠

❝❡r❝❛❞♦ ❝♦♠ ❞✐♠❡♥sõ❡s 4m ① 12m✱ ✉t✐❧✐③❛♥❞♦ ❛♣❡♥❛s✱4m ❞♦ ♠✉r♦✳

❖❜s❡r✈❡ q✉❡ t❡♠♦s ❞✉❛s s♦❧✉çõ❡s ❞✐st✐♥t❛s ♣❛r❛ ❛ ♥♦ss❛ ❡q✉❛çã♦✱ ♠❛s ✉♠❛ ❞❡❧❛s ♥ã♦ é s♦❧✉çã♦ ❞♦ ♥♦ss♦ ♣r♦❜❧❡♠❛✳ ➱ ✐♠♣♦rt❛♥t❡ q✉❡ ♦s ❛❧✉♥♦s ♣❡r❝❡❜❛♠✱ ❛tr❛✈és ❞❡ ♣r♦❜❧❡♠❛s

❈❛♣ít✉❧♦ ✶ Pr♦❜❧❡♠❛s ❊♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ◗✉❛❞rát✐❝❛s

❝♦♠♦ ❡st❡✱ q✉❡ ♥❡♠ s❡♠♣r❡ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ sã♦ ❛s s♦❧✉çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ ❝❛❜❡♥❞♦ ❛ ❡❧❡s ❛♥❛❧✐s❛r ❡ ❞❡❝✐❞✐r q✉❛✐s ❞❡❧❛s s❡ ❛❞❡q✉❛♠ ❛ ❝♦♥❞✐çã♦ ❡st❛❜❡❧❡❝✐❞❛ ✐♥✐❝✐❛❧♠❡♥t❡✳

Pr♦❜❧❡♠❛s q✉❡ ♣♦ss✉❡♠ ♠❛✐s ❞❡ ✉♠❛ s♦❧✉çã♦✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ♥ã♦ sã♦ ❜❡♠ ❡①♣❧♦r❛❞♦s ❡♠ ♥♦ss♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ t❛❧✈❡③ ♣❛r❛ ♥ã♦ ❤❛✈❡r ♠❛✐♦r❡s ❡①♣❧✐❝❛çõ❡s✱ ✈✐st♦ q✉❡ s✐t✉❛çõ❡s ❞❡ss❡ t✐♣♦ ♥❡❝❡ss✐t❛♠ ❞❡ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ❝♦♠♣❧❡①❛ ❡ ❛♣r♦❢✉♥❞❛❞❛✱ ❣❡r❛♥❞♦ ✉♠❛ ♠❛✐♦r ❞✐s✲ ❝✉ssã♦✳ ❊ss❡ t✐♣♦ ❞❡ ❛♥á❧✐s❡ ❛❝❛❜❛ ❞❡s❡♥✈♦❧✈❡♥❞♦✱ ♥♦ ❛❧✉♥♦✱ ✉♠ s❡♥s♦ ❝rít✐❝♦ ❡ ❛✈❛❧✐❛❞♦r✱ q✉❡ ❜✉s❝❛ ❡①❛♠✐♥❛r ❝❛❞❛ s♦❧✉çã♦ ❡♥❝♦♥tr❛❞❛ ❡ ❝♦♠♣❛r❛r ❝♦♠ ❛ r❡❛❧✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛✳

❈❛♣ít✉❧♦ ✷

❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❆♦ ❞❡✜♥✐r♠♦s ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❡st❛r❡♠♦s ❛❞♠✐t✐♥❞♦ q✉❡ ♦s ❛❧✉♥♦s ❥á ❝♦♥❤❡❝❡♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦ ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳

❊❧♦♥ r❡ss❛❧t❛ ❡♠ ❬✺❪ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ t❡r ♦❜❥❡t✐✈✐❞❛❞❡ ❛♦ s❡ ❞❡❢✐♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❢✉♥çã♦✿

❯♠ ❡①❡♠♣❧♦ ❢❧❛❣r❛♥t❡ ❞❛ ❢❛❧t❛ ❞❡ ♦❥❡t✐✈✐❞❛❞❡ ✭q✉❡ ♣❡rs✐st❡ ❛té ❤♦❥❡ ❡♠ q✉❛s❡ t♦❞♦s ♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❜r❛s✐❧❡✐r♦s✮ é ❛ ❞❡❢✐♥✐çã♦ ❞❡ ❢✉♥çã♦ ❝♦♠♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s✳ ❋✉♥çã♦ é ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ♠❛t❡♠át✐❝❛ ✭♦ ♦✉tr♦ é ❝♦♥❥✉♥t♦✮✳ ❖s ✉s✉ár✐♦s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♦s ♣ró♣r✐♦s ♠❛t❡♠át✐❝♦s ❝♦st✉♠❛♠ ♣❡♥s❛r ♥✉♠❛ ❢✉♥çã♦ ❞❡ ♠♦❞♦ ❞✐♥â♠✐❝♦✱ ❡♠ ❝♦♥tr❛st❡ ❝♦♠ ❡ss❛ ❝♦♥✲ ❝❡♣çã♦ ❡stát✐❝❛ ✭✳ ✳ ✳ ✮

P❛r❛ ✉♠ ♠❛t❡♠át✐❝♦✱ ♦✉ ✉♠ ✉s✉ár✐♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ✉♠❛ ❢✉♥çã♦ f : X _→ Y✱ ❝✉❥♦ ♦ ❞♦♠í♥✐♦ é ♦ ❝♦♥❥✉♥t♦ X ❡ ❝♦♥tr❛✲❞♦♠í♥✐♦ ♦ ❝♦♥❥✉♥t♦ Y✱ é ✉♠❛

❝♦rr❡s♣♦♥❞ê♥❝✐❛ ✭✐st♦ é✱ ✉♠❛ r❡❣r❛✱ ✉♠ ❝r✐tér✐♦✱ ✉♠ ❛❧❣♦r✐t♠♦ ♦✉ ✉♠❛ sér✐❡ ❞❡ ✐♥str✉çõ❡s✮ q✉❡ ❡st❛❜❡❧❡❝❡✱ s❡♠ ❡①❝❡çõ❡s ♥❡♠ ❛♠❜✐❣✉✐❞❛❞❡✱ ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦

x❡♠X✱ s✉❛ ✐♠❛❣❡♠f(x)❡♠Y✳

❉❡❢✐♥✐çã♦ ✷✳✶✳ ❈❤❛♠❛r❡♠♦s ❞❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ♦✉ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ❛ ❢✉♥çã♦ f :_R→R✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ x_∈R ♦ ✈❛❧♦r f(x) =ax2_+bx+c

∈R✱ ❝♦♠ a, b

❡ c ♥ú♠❡r♦s r❡❛✐s ❡ a₆= 0✳

❆❧❣✉♠❛s ✈❡③❡s✱ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ✈❛♠♦s ♥♦s r❡❢❡r✐r ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ s✐♠♣❧❡s♠❡♥t❡ ♣♦rf✳

❊①❡♠♣❧♦ ✷✳✶✳

i) ❆ ❢✉♥çã♦ f(x) =x2

−3x+ 7 é ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❝♦♠ a = 1, b=₋3 ❡ c= 7❀

ii) ❆ ❢✉♥çã♦ f(x) =₋9x2_{+ 4x é ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❝♦♠ a =}

−9, b= 4 ❡ c= 0❀

iii) ❆ ❢✉♥çã♦ f(x) = 5x2 _{é ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❝♦♠ a}

= 5, b= 0 ❡ c= 0❀

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

iv) ❆ ❢✉♥çã♦ f(x) = 2x₋17 ♥ã♦ é ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ♣♦✐s a= 0✳

❖ ❢❛t♦ ❞♦ ✈❛❧♦r ❞❡ a s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ❣❛r❛♥t❡ q✉❡ ❡①❡♠♣❧♦s ❝♦♠♦ ♦ (iv)✱ ♥ã♦

s❡❥❛♠ ❝♦♥s✐❞❡r❛❞❛s ❢✉♥çõ❡s q✉❛❞rát✐❝❛s✱ ❞♦ ❝♦♥trár✐♦ t❡rí❛♠♦s ❛ ❢✉♥çã♦ ❛❢✐♠ ❝♦♠♦ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ss❛s ❢✉♥çõ❡s✳

✷✳✶ ❋♦r♠❛ ❈❛♥ô♥✐❝❛

❈♦♥s✐❞❡r❡ ♦ tr✐♥ô♠✐♦ ax2

+bx+c✱ ❝♦♠ a, b ❡c r❡❛✐s ❡a₆= 0✳

❈♦❧♦❝❛♥❞♦ a ❡♠ ❡✈✐❞ê♥❝✐❛ ❡ ✉t✐❧✐③❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦✱ t❡♠♦s✿

ax2+bx+c = a

x2 + b

ax+ c a

= a "

x2+ 2 b 2ax+

b 2a 2 − b 2a 2 + c a # = a " x+ b

2a 2 + c a − b 2a 2# = a

x+ b 2a

2

+c₋ b

2

4a

= a

x+ b 2a

2

+4ac−b

2

4a

P♦❞❡♠♦s ❛ss✐♠✱ r❡❡s❝r❡✈❡r ❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛f(x) = ax2

+bx+c

❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

f(x) =a

x+ b 2a

2

+4ac−b

2

4a . ✭✷✳✶✮

❉❡ ♠❛♥❡✐r❛ ❡q✉✐✈❛❧❡♥t❡✿

f(x) =a(x₋x0)2+y0, ✭✷✳✷✮

♦♥❞❡x0 =−₂b_a ❡y0 = 4ac−b

2

4a ✳

❊st❛ é ❛ ❝❤❛♠❛❞❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ❛ ♣❛rt✐r ❞❛ q✉❛❧✱ ♦❜t❡r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❛f✳

✶❛ _{Pr♦♣r✐❡❞❛❞❡✿ ❱❛❧♦r ♠á①✐♠♦ ❡ ✈❛❧♦r ♠í♥✐♠♦}

❉❡❢✐♥✐çã♦ ✷✳✷✳ ❉❛❞♦ m _{∈ R}✱ ❞✐③❡♠♦s q✉❡ f(m) é ♦ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ❢✉♥çã♦ f s❡ f(x)_≤f(m),_∀x_∈R ❡ ❞✐③❡♠♦s q✉❡ f(m) é ♦ ✈❛❧♦r ♠í♥✐♠♦ s❡f(x)_≥f(m),_∀x_∈R✳

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

◆♦t❡ q✉❡ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ✭✷✳✷✮ ❞❛ ❢✉♥çã♦f é ❝♦♠♣♦st❛ ♣♦r ❞✉❛s ♣❛r❝❡❧❛s✱ ❛ ♣❛r❝❡❧❛✿ a(x₋x0)2 q✉❡ ✈❛r✐❛ ❝♦♠x❡ ❛ ♣❛r❝❡❧❛y0 = 4ac−b

2

4a ✱ ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r ✈❛❧♦r❡s ❝♦♥st❛♥t❡s✳ ❙❡a >0✱ ❡♥tã♦ a(x₋x0)2 ≥0❡

a(x₋x0)2+y0 ≥0 +y0.

❆ss✐♠✿

f(x)_≥y0,

♦✉ s❡❥❛✱ f ❛t✐♥❣❡ ♦ ✈❛❧♦r ♠í♥✐♠♦ y0 = 4ac−b

2

4a q✉❛♥❞♦ x−x0 = 0✱ ♦✉ ❛✐♥❞❛✱ ❡♠ x=x0✳ ❙❡a <0✱ ❡♥tã♦ a(x₋x0)2 ≤0❡

a(x₋x0)2+y0 ≤0 +y0.

❆ss✐♠✿

f(x)_≤y0,

♦✉ s❡❥❛✱ f ❛t✐♥❣❡ ♦ ✈❛❧♦r ♠á①✐♠♦y0 = 4ac−b

2

4a q✉❛♥❞♦ x−x0 = 0✱ ♦✉ ❛✐♥❞❛✱ ❡♠ x=x0✳ ❙❡♥❞♦ ❛ss✐♠✱ ♦ ♣♦♥t♦ ❞♦ ❞♦♠í♥✐♦ x0 = −₂b_a é ♦ ♣♦♥t♦ q✉❡ ♠❛①✐♠✐③❛ ♦✉ ♠✐♥✐♠✐③❛ ❛

❢✉♥çã♦✱ ❞❡♣❡♥❞❡♥❞♦ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❞♦ s✐♥❛❧ ❞❡a✳

❘❡s✉♠✐♥❞♦✿

• ❙❡ a >0❛ ❢✉♥çã♦ ❛❞♠✐t❡ ✉♠ ✈❛❧♦r ♠í♥✐♠♦❀

• ❙❡ a <0❛ ❢✉♥çã♦ ❛❞♠✐t❡ ✉♠ ✈❛❧♦r ♠á①✐♠♦✳

❉❡ss❡ ♠♦❞♦✱ ❡st❛♥❞♦ ❛ ❢✉♥çã♦ ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ ✭❞❡♣❡♥❞❡♥❞♦ ❞♦ s✐♥❛❧ ❞❡a✮ é ♠✉✐t♦ s✐♠♣❧❡s✳ ❱❡❥❛♠♦s ♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✿

❊①❡♠♣❧♦ ✷✳✷✳

❊♥❝♦♥tr❡ ♦ ✈❛❧♦r ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ f(x) =x2

−10x+ 21✳

❙♦❧✉çã♦

❊s❝r❡✈❡♥❞♦ f(x) ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ t❡♠♦s✿

f(x) = (x₋5)2

−25 + 21, ♦✉ s❡❥❛✱ f(x) = (x₋5)2

−4✳ ❖ ✈❛❧♦r ♠í♥✐♠♦ ❞❛ ❢✉♥çã♦ é ✲✹

q✉❡ ♦❝♦rr❡ ♥♦ ♣♦♥t♦ ❞♦ ❞♦♠í♥✐♦x= 5✳

❱❛♠♦s✱ ❛❣♦r❛✱ ♣❡♥s❛r ♥♦ Pr♦❜❧❡♠❛ ✶✳✷✱ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ✉♠ ♣♦✉❝♦ ❞✐❢❡r❡♥t❡✳

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

Pr♦❜❧❡♠❛ ✷✳✶✳ ▼❡✉ ❛✈ô ♣♦ss✉✐ ✉♠ t❡rr❡♥♦ r❡t❛♥❣✉❧❛r ❝♦♠ ❞✐♠❡♥sõ❡s ✷✷♠ ♣♦r ✸✵♠✳ ❙❛❜❡✲s❡ q✉❡ ✉♠ ❞♦s ❧❛❞♦s ❞❡ ♠❡❞✐❞❛ ✷✷♠ ❥á ♣♦ss✉✐ ✉♠ ♠✉r♦ ❝♦♥str✉í❞♦ ❡ ❡❧❡ q✉❡r ✉t✐❧✐③❛r ♣❛rt❡ ❞❡ss❡ ♠✉r♦ ♣❛r❛ ❢❛③❡r ✉♠ ❝❡r❝❛❞♦ r❡t❛♥❣✉❧❛r✳ ❉✐s♣♦♥❞♦ ❞❡ ✷✽♠ ❞❡ t❡❧❛✱ q✉❛✐s sã♦ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ss❡ ❝❡r❝❛❞♦ ♣❛r❛ q✉❡ ❡❧❡ ❝♦♥s✐❣❛ ❛ ♠❛✐♦r ár❡❛ ♣♦ssí✈❡❧✳

❙♦❧✉çã♦

❈♦♥s✐❞❡r❛♥❞♦ x ❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦ ♥ã♦ ♣❛r❛❧❡❧♦ ❛♦ ♠✉r♦✱ ♦ ❧❛❞♦ q✉❡ é ♣❛r❛❧❡❧♦ ♠❡❞❡

(28₋2x)✱ ❞❡ss❛ ❢♦r♠❛ t❡♠♦s q✉❡ ❛ ár❡❛ ✭❡♠ m2_{✮ é ✉♠❛ ❢✉♥çã♦ ❞♦ ❧❛❞♦ x✱ q✉❡ ✈❛♠♦s}

❞❡♥♦t❛r ♣♦r A(x) =x(28₋2x)✳

❊s❝r❡✈❡♥❞♦ ❛ ❡①♣r❡ssã♦A(x) = x(28₋2x)❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ♦❜t❡♠♦s✿

A(x) = ₋2(x₋7)2

+ 98,

❧♦❣♦✱ s✉❛ ár❡❛ ♠á①✐♠❛ s❡rá ❞❡98m2_{✱ s❡♥❞♦ ♣♦ssí✈❡❧ q✉❛♥❞♦ x}

= 7m ❡ ♦ ❧❛❞♦ ♣❛r❛❧❡❧♦ ❛♦

♠✉r♦ ♣♦ss✉✐r14m ❞❡ ❝♦♠♣r✐♠❡♥t♦✳

✷❛ _{Pr♦♣r✐❡❞❛❞❡✿ ❩❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛}

❖s ③❡r♦s ❞❛ ❢✉♥çã♦fsã♦ ♦s ✈❛❧♦r❡s ❞❡x♣❛r❛ ♦s q✉❛✐sf(x) = 0✱ ♦✉ ❞❡ ❢♦r♠❛ ❡q✉✐✈❛❧❡♥t❡

sã♦ ❛s r❛í③❡s✶ _{❞❛ ❡q✉❛çã♦ ax}2_{+bx+c= 0✳}

❖❜s❡r✈❡ q✉❡ ❛ r❡s♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ✶✳✷ é ♥❛ ✈❡r❞❛❞❡ ❛ ❜✉s❝❛ ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦

x2

−14x+ 24 = 0✳

❉❡t❡r♠✐♥❛r ♦s ③❡r♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ❡st❛♥❞♦ ❡❧❛ ❡♠ s✉❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ♥ã♦ é ✉♠❛ t❛r❡❢❛ ❝♦♠♣❧✐❝❛❞❛✳ ❉❡ ❢❛t♦✱ ♣❛rt✐♥❞♦ ❞❡f(x) = 0✱ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦✿

a

x+ b 2a

2

+4ac−b

2

4a = 0, ✭✷✳✸✮

❝✉❥❛ s♦❧✉çã♦ é ❛ ✏❢❛♠♦s❛✑ ❢ór♠✉❧❛✷ _{❛♣r❡s❡♥t❛❞❛ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✿}

x= −b±

√

b2

−4ac

2a ✭✷✳✹✮

✶_{❆ss✐♠ ❝♦♠♦ ❉❛♥✐❡❧ ❈♦r❞❡✐r♦ ❡♠ ❬✽❪✱ ❝❤❛♠❛r❡♠♦s ❞❡ r❛í③❡s ♦s ✈❛❧♦r❡s ❞❡xq✉❡ r❡s♦❧✈❡♠ ✉♠❛ ❡q✉❛çã♦}

❡ ✉s❛r❡♠♦s ♦ t❡r♠♦ ③❡r♦s ♣❛r❛ ❢✉♥çõ❡s✳

✷_{❉❡ ♠❛♥❡✐r❛ ❡q✉✐✈♦❝❛❞❛✱ ❡ss❛ ❡①♣r❡ssã♦ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦}

❜r❛s✐❧❡✐r♦✳ ❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ✈✐♠♦s q✉❡ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s q✉❛❞rát✐❝❛s ♣♦r ♠❡✐♦ ❞❡ ❢ór♠✉❧❛s r❡s♦❧✉t✐✈❛s ❥á ❡r❛ tr❛❜❛❧❤❛❞❛ ❛ q✉❛s❡ ✹✵✵✵ ❛♥♦s ❡ ♦ ♠❛t❡♠át✐❝♦ ➪r❛❜❡ ❇❛s❦❤❛r❛ é ❞♦ sé❝✉❧♦ ✶✷✳ ❙❡♥❞♦ ❛ss✐♠✱ ♥ã♦ ❢❛③ s❡♥t✐❞♦ ❜❛t✐③❛r ❛ ❢ór♠✉❧❛ ❝♦♠ ♦ ♥♦♠❡ ❞❡ss❡ ♠❛t❡♠át✐❝♦✳

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❖ t❡r♠♦ b2

−4ac é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛ ❣r❡❣❛ ∆✭❞❡❧t❛✮✱

∆ =b2₋4ac ✭✷✳✺✮

❡ t❡♠ ✐♠♣♦rtâ♥❝✐❛ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❡st✉❞♦ ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✳ ❈♦♠ ❡ss❛ ♥♦✈❛ ♥♦t❛çã♦✱ ❛ ❡q✉❛çã♦ ✭✷✳✹✮ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r✿

x= −b±

√

∆

2a . ✭✷✳✻✮

◆ã♦ ✈❡♠♦s ♠♦t✐✈♦ ♣❛r❛ q✉❡ ❡ss❛ ❡①♣r❡ssã♦ s❡❥❛ ❛♣r❡s❡♥t❛❞❛ ❛♦s ❛❧✉♥♦s ❝♦♠♦ ❛ ❢ór✲ ♠✉❧❛ ♣❛r❛ ❡♥❝♦♥tr❛r ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✱ ♣♦✐s ♦ ♠ét♦❞♦ ❞❡ ❝♦♠♣❧❡t❛r q✉❛❞r❛❞♦ ❝✉♠♣r❡ ♠✉✐t♦ ❜❡♠ ❡st❛ t❛r❡❢❛✳

❈❛rr❡❣❛r ♦s ❛❧✉♥♦s ❝♦♠ ❢ór♠✉❧❛s q✉❡ ♥ã♦ ♣♦ss✉❛♠ s✐❣♥✐✜❝❛❞♦ ❝♦♥❝❧✉s✐✈♦✱ ❛❝❛❜❛ ♣r✐✲ ✈✐❧❡❣✐❛♥❞♦ ♦ ❛♣r❡♥❞✐③❛❞♦ ♠❡❝â♥✐❝♦✱ q✉❡ ♥ã♦ ❞❡s❡♥✈♦❧✈❡ ♦ r❛❝✐♦❝í♥✐♦ ❞♦ ❛❧✉♥♦ ❡ ❞✐s♣❡r❞✐ç❛ t♦❞♦ ♣♦t❡♥❝✐❛❧ q✉❡ ❡❧❡ ♣♦ss✉✐ ♣❛r❛ ❜✉s❝❛r ♦ ❛♣r❡♥❞✐③❛❞♦✳

❈♦♥t✉❞♦✱ ❛ ♣❛rt✐r ❞❛ á♥❛❧✐s❡ ❞❛ ❢ór♠✉❧❛ ✭✷✳✻✮ ♣♦❞❡♠♦s ❡①tr❛✐r ✐♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ♦ ❡st✉❞♦ ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉✳ ❉❛s ❡q✉❛çõ❡s ✷✳✸ ❡ ✷✳✺ ♦❜t❡♠♦s ❛ ❡q✉❛çã♦✿

x+ b

2a 2

= ∆ 4a2

q✉❡ só ❢t❡♠ s♦❧✉çã♦ q✉❛♥❞♦ ∆_≥0✱ ❥á q✉❡ ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ❡q✉❛çã♦ é ✉♠ ♥ú♠❡r♦

❡❧❡✈❛❞♦ ❛♦ q✉❛❞r❛❞♦✳❉✐③❡♠♦s ❡♥tã♦ q✉❡✿

• ❙❡ ∆_≥0✱ ❛ ❢✉♥çã♦ ♣♦ss✉✐ ③❡r♦s r❡❛✐s❀

• ❈❛s♦ ❝♦♥trár✐♦✱ ♦✉ s❡❥❛✱ ∆<0✱ ❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐ ③❡r♦s r❡❛✐s✳

◗✉❛♥❞♦ ∆ = 0✱ (x+₂b_a)2

= 0 ❡ ♣♦rt❛♥t♦ x=₋₂b_a é ♦ ú♥✐❝♦ ③❡r♦ ❞❛ ❢✉♥çã♦✳

◗✉❛♥❞♦ ❡st❛♠♦s tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❡q✉❛çõ❡s ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡ ♦ ✈❛❧♦r ❞❡∆é ♥❡❣❛t✐✈♦✱

❛ ❡q✉❛çã♦ ♥ã♦ ♣♦ss✉✐ s♦❧✉çã♦ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ◆♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ✈❡r❡♠♦s q✉❡ ♠❡s♠♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐♥❞♦ ③❡r♦s r❡❛✐s✱ ❛✐♥❞❛ é ♣♦ssí✈❡❧✱ ♣♦r ❡①❡♠♣❧♦✱ ❡s❜♦ç❛r s❡✉ ❣rá✜❝♦ ❡ ❞❡t❡r♠✐♥❛r ♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ♦✉ ❞❡ ♠í♥✐♠♦✳

❊①❡♠♣❧♦ ✷✳✸✳ ❉❡t❡r♠✐♥❡ ♦s ③❡r♦s ❞❛s ❢✉♥çõ❡s ❛❜❛✐①♦✿

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

a)f(x) =x2

+ 5x+ 6

❉♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡ q✉❛❞r❛❞♦s t❡♠♦s✿

x2 + 5x+ 6 =

x+ 5

2

2

− 1₄ = 0,

q✉❡ ❡q✉✐✈❛❧❡ ❛✿

x+5

2

2

= 1

4, ♦✉ s❡❥❛✱ x=− 5 2 ±

1 2.

P♦rt❛♥t♦

x=₋2 ♦✉x=₋3.

b)f(x) = 2x2

−3x+ 5

❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦ t❡♠♦s✿

2x2₋3x+ 5 = 2

x₋ 3

4

2

+31 8 = 0,

q✉❡ é ❡q✉✐✈❛❧❡♥t❡ ❛✿

x₋ 3

4

2

=₋31 16.

▲♦❣♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐ ③❡r♦s r❡❛✐s✱ ♣♦✐s ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ é ✉♠ t❡r♠♦ ❡❧❡✈❛❞♦ ❛♦ q✉❛❞r❛❞♦✳

c)f(x) = ₋x2_{+ 2x}

−1

❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦ t❡♠♦s✿

−x2+ 2x₋1 = (₋1)(x₋1)2

= 0,

q✉❡ é ❡q✉✐✈❛❧❡ ❛✿

(x₋1)2 = 0, ♦✉ s❡❥❛✱ x= 1.

◆❡st❡ ❝❛s♦ ✉♠ ú♥✐❝♦ ✈❛❧♦r é ③❡r♦ ❞❡st❛ ❢✉♥çã♦✳

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

✷✳✷ ❋♦r♠❛ ❋❛t♦r❛❞❛

P❛r❛ ❡s❝r❡✈❡r ❛ ❢♦r♠❛ ❢❛t♦r❛❞❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ ♣r❡❝✐s❛♠♦s ❞❛s ❡①♣r❡ssõ❡s ♣❛r❛ ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡♠ ❢✉♥çã♦ ❛♣❡♥❛s ❞❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❈❤❛♠❛♥❞♦ ❛s r❛í③❡s ❞❛ ❡q✉❛çã♦ ❞❡ α = −b−

√

b2

−4ac

2a ❡ β = −

b+√b2

−4ac

2a ❡

❞❡♥♦t❛♥❞♦ ♣♦rs ❛ s♦♠❛ ❞❛s r❛í③❡s ♦❜t❡♠♦s✿

s = −b+

√

b2_−4ac

2a +

−b₋√b2_−4ac

2a

= ₋2b 2a

= ₋b

a. ✭✷✳✼✮

❉❡♥♦t❛♥❞♦ ♣♦r p ♦ ♣r♦❞✉t♦ ❞❛s r❛í③❡s ❡ ♣r♦❝❡❞❡♥❞♦ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✿

p =

−b+√b2_−4ac

2a

.

−b₋√b2_−4ac

2a

= (₋1)

(₋b+√b2

−4ac)(b+√b2

−4ac) 4a2

= (₋1)

−b2+b2₋4ac

4a2

= c

a. ✭✷✳✽✮

❙❡❥❛♠ α ❡ β ❛s r❛í③❡s r❡❛✐s ❞❛ ❡q✉❛çã♦ ax2 _{+bx+c= 0✳ ❈♦❧♦❝❛♥❞♦ a ❡♠ ❡✈✐❞ê♥❝✐❛}

♥❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦ ❞❛ ❢✉♥çã♦ f ❡ ✉s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✷✳✼✮ ❡ ✭✷✳✽✮ ♦❜t❡♠♦s✿

f(x) = a[x2₋(α+β)x+αβ] = a[x2₋αx₋βx+αβ] = a[x(x₋α)₋β(x₋α)]

= a[(x₋α)(x₋β)]. ✭✷✳✾✮

❆ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞♦ s❡❣✉♥❞♦ ❣r❛✉ ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ ✭✷✳✾✮ é ❝❤❛♠❛❞❛ ❢♦r♠❛ ❢❛t♦r❛❞❛✱ ❛ ♣❛rt✐r ❞❛ q✉❛❧ ♦❜t❡r❡♠♦s ♠❛✐s ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞❛ f✿

✸❛ _{Pr♦♣r✐❡❞❛❞❡✿ ❙✐♥❛❧ ❞❛ ❢✉♥çã♦}

❊st✉❞❛r ♦ s✐♥❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ f é ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❞❡ x ♣❛r❛ ♦s q✉❛✐s ❛ ✐♠❛❣❡♠

❈❛♣ít✉❧♦ ✷ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

f(x) é ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦ ❡ ♦s ✈❛❧♦r❡s ❞❡ x ♣❛r❛ ♦s q✉❛✐s ❛ ✐♠❛❣❡♠f(x) é ✉♠ ♥ú♠❡r♦

♣♦s✐t✐✈♦✳

❙❡❥❛♠ α ❡ β ♦s ③❡r♦s ❞❛ ❢✉♥çã♦ f(x) =ax2

+bx+c ❡ ✈❛♠♦s ❛❞♠✐t✐rα < β✳

P❛r❛ ♦s ✈❛❧♦r❡s ❞❡ x q✉❡ ❡stã♦ ❡♥tr❡ ❛s r❛í③❡s ✭α < x < β✮✱ ♦ ♣r♦❞✉t♦ (x₋α)(x₋β)

é ♥❡❣❛t✐✈♦✱ ❛ss✐♠ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✾✮ ♦ s✐♥❛❧ ❞❡ f s❡rá ❝♦♥trár✐♦ ❛♦ ❞❡ a✳

P❛r❛ ♦s ✈❛❧♦r❡s ✭x < α♦✉x > β✮ ❞❡xq✉❡ ❡stã♦ ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❡α❡β✱ ♦ ♣r♦❞✉t♦

(x₋α)(x₋β) é ♣♦s✐t✐✈♦ ❡ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✾✮ ♦ s✐♥❛❧ ❞❡f é ♦ ♠❡s♠♦ ❞❡ a✳

❈❛s♦ αs❡❥❛ ✐❣✉❛❧ ❛β✱ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✾✮ t❡♠♦sf(x) =a(x₋α)2_{✱ ♥❡st❡ ❝❛s♦ ❛ ❢✉♥çã♦}

s❡ ❛♥✉❧❛ ❛♣❡♥❛s ♣❛r❛x=α ❡ t❡rá ♦ ♠❡s♠♦ s✐♥❛❧ ❞❡ a ♣❛r❛ ♦s ❞❡♠❛✐s ✈❛❧♦r❡s r❡❛✐s ❞❡ x✱

✈✐st♦ q✉❡ (x₋α)2 _{é s❡♠♣r❡ ♣♦s✐t✐✈♦✱ s❡x}

6

=α✳

◆♦ ❝❛s♦ ❞❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐r ③❡r♦s r❡❛✐s✱ ♥ã♦ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ❡♠ s✉❛ ❢♦r♠❛ ❢❛t♦r❛❞❛✱ ♣♦ré♠ ♦ ❡st✉❞♦ ❞❡ s❡✉s s✐♥❛✐s ♣♦❞❡ s❡r ❛♥❛❧✐s❛❞♦ ❛tr❛✈és ❞♦ ✈❛❧♦r ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦ q✉❡ ❛ ❢✉♥çã♦ ❛ss✉♠❡✳ ❱❡❥❛♠♦s✿

❉❛ s❡çã♦ ✷✳✶ s❛❜❡♠♦s q✉❡ s❡ a >0 ❛ ❢✉♥çã♦ ♣♦ss✉✐ ✈❛❧♦r ♠í♥✐♠♦₋∆

4a✱ ❝♦♠♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐ ③❡r♦s✱ ♦✉ s❡❥❛✱∆<0✱ ❡ss❡ s❡✉ ✈❛❧♦r ♠í♥✐♠♦ s❡rá ♣♦s✐t✐✈♦ ❡ ♣♦rt❛♥t♦f(x)_≥0✱

♣❛r❛ t♦❞♦x_∈R✳

❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ s❡ a <0 ❛ ❢✉♥çã♦ ♣♦ss✉✐ ✈❛❧♦r ♠á①✐♠♦ ₋∆

4a✱ ❝♦♠♦ ❛ ❢✉♥çã♦ ♥ã♦ ♣♦ss✉✐ ③❡r♦s✱ ♦✉ s❡❥❛✱ ∆ < 0✱ ❡ss❡ s❡✉ ✈❛❧♦r ♠á①✐♠♦ s❡rá ♥❡❣❛t✐✈♦ ❡ ♣♦rt❛♥t♦ f(x) _≤ 0✱

♣❛r❛ t♦❞♦x_∈R✳

❈❛♣ít✉❧♦ ✸

●rá❢✐❝♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

◆♦ ❡♥s✐♥♦ ❞❡ ❢✉♥çõ❡s q✉❛❞rát✐❝❛s é ❝♦♠✉♠ ♦ ♣r♦❢❡ss♦r ❝♦♥str✉✐r ✉♠❛ t❛❜❡❧❛ ❝♦♠ ❛❧❣✉♥s ✈❛❧♦r❡s ♣❛r❛ x✱ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ♣❛r❛ f(x)✱ ♠❛r❝❛r ❡ss❡s ♣♦♥t♦s ♥♦

♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✱ ✉♥✐✲❧♦s s❡❣✉✐♥❞♦ ♦s tr❛ç♦s ❞❡ ✉♠❛ ♣❛rá❜♦❧❛ ❡ ❡♥✉♥❝✐❛r✿ ✏❱❡❥❛♠ ❡ss❡ ❣rá✜❝♦ r❡♣r❡s❡♥t❛ ✉♠❛ ♣❛rá❜♦❧❛✑✳

❊ ❛ss✐♠ é ❞❛❞❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣❛rá❜♦❧❛ ♣❛r❛ ♦s ❛❧✉♥♦s✿ ✏P❛rá❜♦❧❛ é ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✑

❈♦♠ ❡ss❛ ❞❡✜♥✐çã♦ ♦ ❛❧✉♥♦ ❛❝❛❜❛ ❛ss♦❝✐❛♥❞♦ ❛ ♣❛rá❜♦❧❛✱ ❞❡ ❢♦r♠❛ ❡q✉✐✈♦❝❛❞❛✱ ❛ q✉❛❧q✉❡r ❣rá✜❝♦ ♦✉ ✜❣✉r❛ q✉❡ ♣♦ss✉❛ ♦ ❢♦r♠❛t♦ s✐♠✐❧❛r ❛♦ ❞❡❧❛✳

P❛r❛ ❡✈✐t❛r ❝♦♥❢✉sõ❡s ❞❡ss❡ t✐♣♦ é ✐♠♣♦rt❛♥t❡ ❞❡✜♥✐r ❝♦rr❡t❛♠❡♥t❡ ❛ ♣❛rá❜♦❧❛ ❡ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡ ♠♦str❛r q✉❡ ❡ss❡s ❝♦♥❥✉♥t♦s sã♦ ✐❣✉❛✐s✳

◆❡st❡ ❝❛♣ít✉❧♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ❡①♣❧✐❝❛r ♦ s✐❣♥✐✜❝❛❞♦ ♠❛t❡♠át✐❝♦ ❞❛ ❢r❛s❡✿ ❖ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ é ✉♠❛ ♣❛rá❜♦❧❛✳

■♥✐❝✐❛❧♠❡♥t❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❞❡ ♣❛rá❜♦❧❛ ❡ ❞❡ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ q✉❛❞rá✲ t✐❝❛ ✳

❉❡❢✐♥✐çã♦ ✸✳✶✳ ❉❛❞❛ ✉♠❛ r❡t❛ d ❝❤❛♠❛❞❛ ❞❡ ❞✐r❡tr✐③ ❡ ✉♠ ♣♦♥t♦ F✱ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡

❛ r❡t❛ ❞❛❞❛✱ ❝❤❛♠❛❞♦ ❞❡ ❢♦❝♦✱ ❝❤❛♠❛♠♦s ❞❡ ♣❛rá❜♦❧❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞♦ ❢♦❝♦ F ❡ ❞❛ ❞✐r❡tr✐③ d✳

❆ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r à ❞✐r❡tr✐③ ♣❛ss❛♥❞♦ ♣♦rF s❡rá ❝❤❛♠❛❞❛ ❞❡ ❡✐①♦ ❞❛ ♣❛rá❜♦❧❛ ❡ ♦

♣♦♥t♦ V✱ q✉❡ é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ ❝♦♠ ❡①tr❡♠✐❞❛❞❡s ❡♠F ❡ ♥❛ ✐♥t❡rs❡❝çã♦ ❞❛

❞✐r❡tr✐③ ❝♦♠ ♦ ❡✐①♦ ❞❛ ♣❛rá❜♦❧❛✱ s❡rá ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡✳

❈❛♣ít✉❧♦ ✸ ●rá❢✐❝♦ ❞❛ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❱❛♠♦s ✉t✐❧✐③❛r ✉♠ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛ ♣❛r❛ ✈✐s✉❛❧✐③❛r ❛ ❝♦♥str✉çã♦ ❞❛ ♣❛rá❜♦❧❛ ♣❛ss♦ ❛ ♣❛ss♦ ❛ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦✳

Pr✐♠❡✐r♦ ♣❛ss♦✿

▼❛r❝❛r ❞♦✐s ♣♦♥t♦s✱A ❡B✱ ♥♦ ♣❧❛♥♦ ♣❛r❛ ❞❡✜♥✐r ❛ r❡t❛ ❞✐r❡tr✐③d✳ ❊♠ s❡❣✉✐❞❛ ♠❛r❝❛r

✉♠ ♣♦♥t♦ q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ r❡t❛✱ ❡ss❡ s❡rá ♥♦ss♦ ❢♦❝♦F✱ ♣❛r❛ ✐ss♦ ✉s❛r❡♠♦s ❛ ❢❡rr❛♠❡♥t❛

◆♦✈♦ P♦♥t♦ ✳

❙❡❣✉♥❞♦ ♣❛ss♦✿

❯s❛♥❞♦ ❛ ❢❡rr❛♠❡♥t❛ ❘❡t❛ ❞❡✜♥✐❞❛ ♣♦r ❉♦✐s P♦♥t♦s ❡ s❡❧❡❝✐♦♥❛♥❞♦ ♦s ♣♦♥t♦s

A ❡B ❝♦♥tr✉í♠♦s ❛ r❡t❛ ❞✐r❡tr✐③ d✳