y uchun yechish
y=\frac{1}{2}=0,5
y=1
Grafik
Baham ko'rish
Klipbordga nusxa olish
9y^{2}-12y+4-y^{2}=0
Ikkala tarafdan y^{2} ni ayirish.
8y^{2}-12y+4=0
8y^{2} ni olish uchun 9y^{2} va -y^{2} ni birlashtirish.
2y^{2}-3y+1=0
Ikki tarafini 4 ga bo‘ling.
a+b=-3 ab=2\times 1=2
Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2y^{2}+ay+by+1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.
a=-2 b=-1
ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. Faqat bundan juftlik tizim yechimidir.
\left(2y^{2}-2y\right)+\left(-y+1\right)
2y^{2}-3y+1 ni \left(2y^{2}-2y\right)+\left(-y+1\right) sifatida qaytadan yozish.
2y\left(y-1\right)-\left(y-1\right)
Birinchi guruhda 2y ni va ikkinchi guruhda -1 ni faktordan chiqaring.
\left(y-1\right)\left(2y-1\right)
Distributiv funktsiyasidan foydalangan holda y-1 umumiy terminini chiqaring.
y=1 y=\frac{1}{2}
Tenglamani yechish uchun y-1=0 va 2y-1=0 ni yeching.
9y^{2}-12y+4-y^{2}=0
Ikkala tarafdan y^{2} ni ayirish.
8y^{2}-12y+4=0
8y^{2} ni olish uchun 9y^{2} va -y^{2} ni birlashtirish.
y=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\times 4}}{2\times 8}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -12 ni b va 4 ni c bilan almashtiring.
y=\frac{-\left(-12\right)±\sqrt{144-4\times 8\times 4}}{2\times 8}
-12 kvadratini chiqarish.
y=\frac{-\left(-12\right)±\sqrt{144-32\times 4}}{2\times 8}
-4 ni 8 marotabaga ko'paytirish.
y=\frac{-\left(-12\right)±\sqrt{144-128}}{2\times 8}
-32 ni 4 marotabaga ko'paytirish.
y=\frac{-\left(-12\right)±\sqrt{16}}{2\times 8}
144 ni -128 ga qo'shish.
y=\frac{-\left(-12\right)±4}{2\times 8}
16 ning kvadrat ildizini chiqarish.
y=\frac{12±4}{2\times 8}
-12 ning teskarisi 12 ga teng.
y=\frac{12±4}{16}
2 ni 8 marotabaga ko'paytirish.
y=\frac{16}{16}
y=\frac{12±4}{16} tenglamasini yeching, bunda ± musbat. 12 ni 4 ga qo'shish.
y=1
16 ni 16 ga bo'lish.
y=\frac{8}{16}
y=\frac{12±4}{16} tenglamasini yeching, bunda ± manfiy. 12 dan 4 ni ayirish.
y=\frac{1}{2}
\frac{8}{16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.
y=1 y=\frac{1}{2}
Tenglama yechildi.
9y^{2}-12y+4-y^{2}=0
Ikkala tarafdan y^{2} ni ayirish.
8y^{2}-12y+4=0
8y^{2} ni olish uchun 9y^{2} va -y^{2} ni birlashtirish.
8y^{2}-12y=-4
Ikkala tarafdan 4 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.
\frac{8y^{2}-12y}{8}=-\frac{4}{8}
Ikki tarafini 8 ga bo‘ling.
y^{2}+\left(-\frac{12}{8}\right)y=-\frac{4}{8}
8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.
y^{2}-\frac{3}{2}y=-\frac{4}{8}
\frac{-12}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.
y^{2}-\frac{3}{2}y=-\frac{1}{2}
\frac{-4}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.
y^{2}-\frac{3}{2}y+\left(-\frac{3}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{3}{4}\right)^{2}
-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
y^{2}-\frac{3}{2}y+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.
y^{2}-\frac{3}{2}y+\frac{9}{16}=\frac{1}{16}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(y-\frac{3}{4}\right)^{2}=\frac{1}{16}
y^{2}-\frac{3}{2}y+\frac{9}{16} 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{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
y-\frac{3}{4}=\frac{1}{4} y-\frac{3}{4}=-\frac{1}{4}
Qisqartirish.
y=1 y=\frac{1}{2}
\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.
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