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\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)
y qiymati -\frac{5}{2},\frac{2}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3y-2\right)\left(2y+5\right) ga, 2y+5,-3y+2 ning eng kichik karralisiga ko‘paytiring.
24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)
3y-2 ga 8y-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)
5 ga -5-2y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.
24y^{2}-31y+10=-145y-175-30y^{2}
-25-10y ga 3y+7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
24y^{2}-31y+10+145y=-175-30y^{2}
145y ni ikki tarafga qo’shing.
24y^{2}+114y+10=-175-30y^{2}
114y ni olish uchun -31y va 145y ni birlashtirish.
24y^{2}+114y+10-\left(-175\right)=-30y^{2}
Ikkala tarafdan -175 ni ayirish.
24y^{2}+114y+10+175=-30y^{2}
-175 ning teskarisi 175 ga teng.
24y^{2}+114y+10+175+30y^{2}=0
30y^{2} ni ikki tarafga qo’shing.
24y^{2}+114y+185+30y^{2}=0
185 olish uchun 10 va 175'ni qo'shing.
54y^{2}+114y+185=0
54y^{2} ni olish uchun 24y^{2} va 30y^{2} ni birlashtirish.
y=\frac{-114±\sqrt{114^{2}-4\times 54\times 185}}{2\times 54}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 54 ni a, 114 ni b va 185 ni c bilan almashtiring.
y=\frac{-114±\sqrt{12996-4\times 54\times 185}}{2\times 54}
114 kvadratini chiqarish.
y=\frac{-114±\sqrt{12996-216\times 185}}{2\times 54}
-4 ni 54 marotabaga ko'paytirish.
y=\frac{-114±\sqrt{12996-39960}}{2\times 54}
-216 ni 185 marotabaga ko'paytirish.
y=\frac{-114±\sqrt{-26964}}{2\times 54}
12996 ni -39960 ga qo'shish.
y=\frac{-114±6\sqrt{749}i}{2\times 54}
-26964 ning kvadrat ildizini chiqarish.
y=\frac{-114±6\sqrt{749}i}{108}
2 ni 54 marotabaga ko'paytirish.
y=\frac{-114+6\sqrt{749}i}{108}
y=\frac{-114±6\sqrt{749}i}{108} tenglamasini yeching, bunda ± musbat. -114 ni 6i\sqrt{749} ga qo'shish.
y=\frac{-19+\sqrt{749}i}{18}
-114+6i\sqrt{749} ni 108 ga bo'lish.
y=\frac{-6\sqrt{749}i-114}{108}
y=\frac{-114±6\sqrt{749}i}{108} tenglamasini yeching, bunda ± manfiy. -114 dan 6i\sqrt{749} ni ayirish.
y=\frac{-\sqrt{749}i-19}{18}
-114-6i\sqrt{749} ni 108 ga bo'lish.
y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}
Tenglama yechildi.
\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)
y qiymati -\frac{5}{2},\frac{2}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3y-2\right)\left(2y+5\right) ga, 2y+5,-3y+2 ning eng kichik karralisiga ko‘paytiring.
24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)
3y-2 ga 8y-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)
5 ga -5-2y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.
24y^{2}-31y+10=-145y-175-30y^{2}
-25-10y ga 3y+7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
24y^{2}-31y+10+145y=-175-30y^{2}
145y ni ikki tarafga qo’shing.
24y^{2}+114y+10=-175-30y^{2}
114y ni olish uchun -31y va 145y ni birlashtirish.
24y^{2}+114y+10+30y^{2}=-175
30y^{2} ni ikki tarafga qo’shing.
54y^{2}+114y+10=-175
54y^{2} ni olish uchun 24y^{2} va 30y^{2} ni birlashtirish.
54y^{2}+114y=-175-10
Ikkala tarafdan 10 ni ayirish.
54y^{2}+114y=-185
-185 olish uchun -175 dan 10 ni ayirish.
\frac{54y^{2}+114y}{54}=-\frac{185}{54}
Ikki tarafini 54 ga bo‘ling.
y^{2}+\frac{114}{54}y=-\frac{185}{54}
54 ga bo'lish 54 ga ko'paytirishni bekor qiladi.
y^{2}+\frac{19}{9}y=-\frac{185}{54}
\frac{114}{54} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.
y^{2}+\frac{19}{9}y+\left(\frac{19}{18}\right)^{2}=-\frac{185}{54}+\left(\frac{19}{18}\right)^{2}
\frac{19}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{19}{18} olish uchun. Keyin, \frac{19}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{185}{54}+\frac{361}{324}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{19}{18} kvadratini chiqarish.
y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{749}{324}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{185}{54} ni \frac{361}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(y+\frac{19}{18}\right)^{2}=-\frac{749}{324}
y^{2}+\frac{19}{9}y+\frac{361}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.
\sqrt{\left(y+\frac{19}{18}\right)^{2}}=\sqrt{-\frac{749}{324}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
y+\frac{19}{18}=\frac{\sqrt{749}i}{18} y+\frac{19}{18}=-\frac{\sqrt{749}i}{18}
Qisqartirish.
y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}
Tenglamaning ikkala tarafidan \frac{19}{18} ni ayirish.