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5x-2\left(x-1\right)\left(3-x\right)-11=0
Ikkala tarafdan 11 ni ayirish.
5x+\left(-2x+2\right)\left(3-x\right)-11=0
-2 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.
5x-8x+2x^{2}+6-11=0
-2x+2 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
-3x+2x^{2}+6-11=0
-3x ni olish uchun 5x va -8x ni birlashtirish.
-3x+2x^{2}-5=0
-5 olish uchun 6 dan 11 ni ayirish.
2x^{2}-3x-5=0
ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -5 ni c bilan almashtiring.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}
-3 kvadratini chiqarish.
x=\frac{-\left(-3\right)±\sqrt{9-8\left(-5\right)}}{2\times 2}
-4 ni 2 marotabaga ko'paytirish.
x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\times 2}
-8 ni -5 marotabaga ko'paytirish.
x=\frac{-\left(-3\right)±\sqrt{49}}{2\times 2}
9 ni 40 ga qo'shish.
x=\frac{-\left(-3\right)±7}{2\times 2}
49 ning kvadrat ildizini chiqarish.
x=\frac{3±7}{2\times 2}
-3 ning teskarisi 3 ga teng.
x=\frac{3±7}{4}
2 ni 2 marotabaga ko'paytirish.
x=\frac{10}{4}
x=\frac{3±7}{4} tenglamasini yeching, bunda ± musbat. 3 ni 7 ga qo'shish.
x=\frac{5}{2}
\frac{10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.
x=-\frac{4}{4}
x=\frac{3±7}{4} tenglamasini yeching, bunda ± manfiy. 3 dan 7 ni ayirish.
x=-1
-4 ni 4 ga bo'lish.
x=\frac{5}{2} x=-1
Tenglama yechildi.
5x-2\left(x-1\right)\left(3-x\right)=11
-2 hosil qilish uchun -1 va 2 ni ko'paytirish.
5x+\left(-2x+2\right)\left(3-x\right)=11
-2 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.
5x-8x+2x^{2}+6=11
-2x+2 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.
-3x+2x^{2}+6=11
-3x ni olish uchun 5x va -8x ni birlashtirish.
-3x+2x^{2}=11-6
Ikkala tarafdan 6 ni ayirish.
-3x+2x^{2}=5
5 olish uchun 11 dan 6 ni ayirish.
2x^{2}-3x=5
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{2x^{2}-3x}{2}=\frac{5}{2}
Ikki tarafini 2 ga bo‘ling.
x^{2}-\frac{3}{2}x=\frac{5}{2}
2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.
x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{5}{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.
x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.
x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(x-\frac{3}{4}\right)^{2}=\frac{49}{16}
x^{2}-\frac{3}{2}x+\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(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-\frac{3}{4}=\frac{7}{4} x-\frac{3}{4}=-\frac{7}{4}
Qisqartirish.
x=\frac{5}{2} x=-1
\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.