Kalkuleta na Tsarin Lissafi Mai Girma Biyu
Warware tsarin lissafi mai girma biyu ax² + bx + c = 0 tare da cikakkun matakai na warwarewa da kuma nazarin hoto
Yadda Ake Amfani da Kalkuleta na Tsarin Lissafi Mai Girma Biyu
- Shigar da ma'aunai a, b, da c na tsarin lissafinka mai girma biyu ax² + bx + c = 0
- Ka lura cewa ma'auni 'a' ba zai iya zama sifili ba (in ba haka ba, ba tsarin lissafi mai girma biyu bane)
- Yi amfani da maɓallan misalai don gwada nau'ikan tsarin lissafi mai girma biyu daban-daban
- Duba tsarin lissafin da ke bayyana kai tsaye don ganin an tsara shi yadda ya kamata
- Duba mai bambancewa don fahimtar nau'in amsoshin da za a sa ran
- Duba warwarewa mataki-mataki don fahimtar yadda ake warwarewa
- Yi nazarin tsayi da layin daidaito don fahimtar hoto
Fahimtar Tsarin Lissafi Mai Girma Biyu
Tsarin lissafi mai girma biyu shine tsarin lissafi na polynomial mai girma 2, wanda aka rubuta a siffa ta yau da kullun ax² + bx + c = 0, inda a ≠ 0.
Ma'auni 'a'
Ma'aunin x². Yana tantance ko parabola tana buɗewa sama (a > 0) ko ƙasa (a < 0).
Importance: Ba zai iya zama sifili ba. Mafi girman |a| yana sa parabola ta zama kankanta.
Ma'auni 'b'
Ma'aunin x. Yana shafar matsayin kwance na tsayi da layin daidaito.
Importance: Zai iya zama sifili. Tare da 'a', yana tantance matsayin x na tsayi: x = -b/(2a).
Ma'auni 'c'
Lambar da ba ta canzawa. Tana wakiltar matsayin y na parabola (inda take ratsa layin y).
Importance: Zai iya zama sifili. Wurin (0, c) shine inda parabola ke ratsa layin y.
Dabarar Tsarin Lissafi Mai Girma Biyu
Dabarar tsarin lissafi mai girma biyu hanya ce ta duniya don warware kowane tsarin lissafi mai girma biyu ax² + bx + c = 0.
Δ = b² - 4ac
x = (-b ± √(b² - 4ac)) / (2a)
Discriminant: Δ = b² - 4ac
Mai bambancewa (Δ) yana tantance yanayi da adadin amsoshi
-b
Kishiyar ma'auni b
Purpose: Yana sa amsoshin su kasance a tsakiyar layin daidaito
±√Δ
Tara/ragi tushen murabba'i na mai bambancewa
Purpose: Yana tantance nisan amsoshin daga tsakiya
2a
Sau biyu na babban ma'auni
Purpose: Yana daidaita amsoshin bisa ga faɗin parabola
Fahimtar Mai Bambancewa
Mai bambancewa Δ = b² - 4ac yana gaya mana game da yanayin amsoshin kafin mu kirga su.
Δ > 0
Sakamako: Amsoshi biyu daban-daban na hakika
Parabola tana ratsa layin x a wurare biyu. Amsoshin lambobi ne na hakika.
Misali: x² - 5x + 6 = 0 yana da Δ = 25 - 24 = 1 > 0, don haka akwai amsoshi biyu na hakika.
A hoto: Parabola tana ratsa layin x sau biyu
Δ = 0
Sakamako: Amsa daya ta hakika da ke maimaita kanta
Parabola tana taɓa layin x a wuri daya kacal (tsayi yana kan layin x).
Misali: x² - 4x + 4 = 0 yana da Δ = 16 - 16 = 0, don haka akwai amsa daya da ke maimaita kanta x = 2.
A hoto: Parabola tana taɓa layin x a tsayi
Δ < 0
Sakamako: Amsoshi biyu masu rikitarwa
Parabola ba ta ratsa layin x. Amsoshin sun haɗa da lambobi na tunani.
Misali: x² + 2x + 5 = 0 yana da Δ = 4 - 20 = -16 < 0, don haka akwai amsoshi masu rikitarwa.
A hoto: Parabola ba ta ratsa layin x
Hanyoyin Warware Tsarin Lissafi Mai Girma Biyu
Dabarar Tsarin Lissafi Mai Girma Biyu
Lokacin amfani: Koyaushe tana aiki ga kowane tsarin lissafi mai girma biyu
Matakai:
- Gano a, b, c
- Kirga mai bambancewa Δ = b² - 4ac
- Yi amfani da dabarar x = (-b ± √Δ)/(2a)
Amfani: Hanya ta duniya, tana nuna mai bambancewa
Rashin Amfani: Zai iya haɗawa da lissafi mai rikitarwa
Rarrabawa
Lokacin amfani: Lokacin da za a iya rarraba tsarin lissafin cikin sauƙi
Matakai:
- Rarraba ax² + bx + c zuwa (px + q)(rx + s)
- Saita kowane bangare zuwa sifili
- Warware px + q = 0 da rx + s = 0
Amfani: Mai sauri lokacin da rarrabawar ta fito fili
Rashin Amfani: Ba duk tsarin lissafi mai girma biyu bane ke rarrabuwa cikin sauƙi
Cika Murabba'i
Lokacin amfani: Lokacin canzawa zuwa siffar tsayi ko fitar da dabarar tsarin lissafi mai girma biyu
Matakai:
- Sake tsara zuwa x² + (b/a)x = -c/a
- Ƙara (b/2a)² zuwa bangarorin biyu
- Rarraba bangaren hagu a matsayin cikakken murabba'i
Amfani: Yana nuna siffar tsayi, mai kyau don fahimta
Rashin Amfani: Matakai da yawa fiye da dabarar tsarin lissafi mai girma biyu
Zane
Lokacin amfani: Don fahimtar gani ko amsoshi na kusa
Matakai:
- Zana parabola y = ax² + bx + c
- Nemo wuraren da ke ratsa layin x inda y = 0
- Karanta amsoshin daga hoton
Amfani: Na gani, yana nuna dukkan siffofi
Rashin Amfani: Wataƙila ba zai ba da ainihin darajoji ba
Aikace-aikacen Tsarin Lissafi Mai Girma Biyu a Rayuwar Yau da Kullun
Fizika - Motsin Abu da Aka Harba
Tsayin abubuwan da aka harba yana bin tsarin lissafi mai girma biyu
Tsarin Lissafi: h(t) = -16t² + v₀t + h₀
Masu Canji: h = tsayi, t = lokaci, v₀ = saurin farko, h₀ = tsayin farko
Matsala: Yaushe abu zai bugi ƙasa? (warware ga t lokacin da h = 0)
Kasuwanci - Inganta Riba
Kudin shiga da riba galibi suna bin tsarin lissafi mai girma biyu
Tsarin Lissafi: P(x) = -ax² + bx - c
Masu Canji: P = riba, x = adadin da aka siyar, ma'aunai sun dogara da farashi
Matsala: Nemo adadin da zai kara riba (tsayin parabola)
Injiniyanci - Tsarin Gada
Lanƙwasa na parabola suna rarraba nauyi yadda ya kamata
Tsarin Lissafi: y = ax² + bx + c
Masu Canji: Yana bayyana lanƙwasar igiyoyin gadar rataye
Matsala: Tsara siffar igiya don ingantaccen rarraba nauyi
Noma - Inganta Wuri
Ƙara faɗin wuri da kewaye da aka sani
Tsarin Lissafi: A = x(P - 2x)/2 = -x² + (P/2)x
Masu Canji: A = faɗin wuri, x = faɗi, P = shingen da ke akwai
Matsala: Nemo ma'aunai da za su kara faɗin wurin da aka kewaye
Fasaha - Sarrafa Sigina
Tsarin lissafi mai girma biyu a cikin matatun dijital da tsarin eriya
Tsarin Lissafi: Siffofi daban-daban dangane da aikace-aikace
Masu Canji: Amsar mita, ƙarfin sigina, lokaci
Matsala: Inganta ingancin sigina da rage tsangwama
Magani - Yawan Magani
Matsayin magani a cikin jini a tsawon lokaci
Tsarin Lissafi: C(t) = -at² + bt + c
Masu Canji: C = yawa, t = lokaci bayan an sha magani
Matsala: Tabbatar da mafi kyawun tazarar shan magani
Kura-kurai da Aka Saba Yi Wajen Warware Tsarin Lissafi Mai Girma Biyu
KUSKURE: Mancewa da ± a cikin dabarar tsarin lissafi mai girma biyu
Matsala: Neman amsa daya kacal alhali akwai biyu
Amsa: Koyaushe a haɗa duka + da - lokacin da mai bambancewa ya fi > 0
Misali: Ga x² - 5x + 6 = 0, duka x = 2 da x = 3 amsoshi ne
KUSKURE: Sanya a = 0
Matsala: Tsarin lissafin ya zama na layi, ba mai girma biyu ba
Amsa: Tabbatar cewa ma'aunin x² ba sifili bane ga tsarin lissafi mai girma biyu
Misali: 0x² + 3x + 2 = 0 a zahiri shine 3x + 2 = 0, tsarin lissafi na layi
KUSKURE: Kura-kuran lissafi da lambobi marasa kyau
Matsala: Kura-kuran alama yayin kirga mai bambancewa ko amfani da dabara
Amsa: A kula da alamun ragi, musamman da b² da -4ac
Misali: Ga x² - 6x + 9, mai bambancewa shine (-6)² - 4(1)(9) = 36 - 36 = 0
KUSKURE: Fassarar amsoshi masu rikitarwa da ba daidai ba
Matsala: Tunani cewa tsarin lissafin ba shi da amsa lokacin da mai bambancewa ya kasa < 0
Amsa: Amsoshi masu rikitarwa suna da inganci a lissafi, kawai ba lambobi na hakika bane
Misali: x² + 1 = 0 yana da amsoshi x = ±i, waɗanda lambobi ne masu rikitarwa
KUSKURE: Jerin ayyuka da ba daidai ba
Matsala: Kirga mai bambancewa da ba daidai ba
Amsa: Ka tuna b² - 4ac: fara da murabba'in b, sannan ka cire 4ac
Misali: Ga 2x² + 3x + 1, mai bambancewa shine 3² - 4(2)(1) = 9 - 8 = 1
KUSKURE: Zagayawa da wuri
Matsala: Kura-kuran zagayawa da aka tara a cikin lissafi mai matakai da yawa
Amsa: A kiyaye cikakken daidaito har zuwa amsar ƙarshe, sannan a zagaya yadda ya kamata
Misali: Yi amfani da cikakken darajar mai bambancewa a cikin dabarar tsarin lissafi mai girma biyu, ba sigar da aka zagaya ba
Yanayi na Musamman da Tsare-tsare
Cikakkun Murabba'ai na Uku
Siffa: a²x² ± 2abx + b² = (ax ± b)²
Misali: x² - 6x + 9 = (x - 3)²
Amsa: Tushe daya da ke maimaita kanta: x = 3
Ganewa: Mai bambancewa daidai yake da sifili
Bambancin Murabba'ai
Siffa: a²x² - b² = (ax - b)(ax + b)
Misali: x² - 16 = (x - 4)(x + 4)
Amsa: Tushe biyu masu kishiyantar juna: x = ±4
Ganewa: Babu kalmar layi (b = 0), lambar da ba ta canzawa mara kyau
Kalmar Layi da ta Bace
Siffa: ax² + c = 0
Misali: 2x² - 8 = 0
Amsa: x² = 4, don haka x = ±2
Ganewa: Kawai kalmomin x² da waɗanda ba sa canzawa ne ke nan
Kalmar da Ba ta Canzawa da ta Bace
Siffa: ax² + bx = 0 = x(ax + b)
Misali: 3x² - 6x = 0 = 3x(x - 2)
Amsa: x = 0 ko x = 2
Ganewa: Fara da rarraba x
Tambayoyi da Amsoshi kan Tsarin Lissafi Mai Girma Biyu
Me ke sa tsarin lissafi ya zama mai girma biyu?
Tsarin lissafi yana da girma biyu idan mafi girman iko na mai canji shine 2, kuma ma'aunin x² ba sifili bane. Dole ne ya kasance a siffar ax² + bx + c = 0.
Shin tsarin lissafi mai girma biyu zai iya zama ba shi da amsa?
Tsarin lissafi mai girma biyu koyaushe yana da amsoshi 2 daidai, amma suna iya zama lambobi masu rikitarwa lokacin da mai bambancewa ya kasance mara kyau. A cikin lambobi na hakika, babu amsoshi lokacin da Δ < 0.
Me yasa wani lokaci muke samun amsa daya maimakon biyu?
Lokacin da mai bambancewa = 0, muna samun amsa daya da ke maimaita kanta (wanda ake kira tushe biyu). A lissafi, har yanzu amsoshi biyu ne da suka zama iri daya.
Me mai bambancewa ke gaya mana?
Mai bambancewa (b² - 4ac) yana tantance nau'ikan amsoshi: mai kyau = amsoshi biyu na hakika, sifili = amsa daya da ke maimaita kanta, mara kyau = amsoshi biyu masu rikitarwa.
Ta yaya zan san wace hanya zan yi amfani da ita?
Dabarar tsarin lissafi mai girma biyu koyaushe tana aiki. Yi amfani da rarrabawa idan tsarin lissafin yana da sauƙin rarrabawa. Yi amfani da cika murabba'i don fahimta ko canzawa zuwa siffar tsayi.
Idan ma'auni na 'a' mara kyau ne fa?
Babu matsala! Dabarar tsarin lissafi mai girma biyu tana aiki da ma'aunai marasa kyau. Kawai a kula da alamomi yayin kirga mai bambancewa da amfani da dabarar.
Shin zan iya warware tsarin lissafi mai girma biyu ba tare da dabarar ba?
I! Zaka iya rarrabawa (idan zai yiwu), cika murabba'i, ko zana hoto. Koyaya, dabarar tsarin lissafi mai girma biyu ita ce hanya mafi inganci ta duniya.
Don me ake amfani da amsoshi masu rikitarwa?
Amsoshi masu rikitarwa suna bayyana a fannin injiniyanci, fizika, da kuma lissafi mai zurfi. Suna wakiltar muhimman alaƙoƙin lissafi koda kuwa ba 'na hakika' bane a ma'anar yau da kullun.
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