7(x-1)(x+1)=(x+1)^2 eching

$$7 ( x - 1 ) ( x + 1 ) = ( x + 1 ) ^ { 2 }$$

x uchun yechish

$$x=-1$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1,333333333$$

Guruhlash orqali faktorlash qadamlari

7 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(7x-7\right)\left(x+1\right)=\left(x+1\right)^{2}$$

7x-7 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$7x^{2}-7=\left(x+1\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$7x^{2}-7=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$7x^{2}-7-x^{2}=2x+1$$

6x^{2} ni olish uchun 7x^{2} va -x^{2} ni birlashtirish.

$$6x^{2}-7=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$6x^{2}-7-2x=1$$

Ikkala tarafdan 1 ni ayirish.

$$6x^{2}-7-2x-1=0$$

-8 olish uchun -7 dan 1 ni ayirish.

$$6x^{2}-8-2x=0$$

Ikki tarafini 2 ga bo‘ling.

$$3x^{2}-4-x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$3x^{2}-x-4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=3\left(-4\right)=-12$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-12 2,-6 3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$1-12=-11 2-6=-4 3-4=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=-4 b=3$$

3x^{2}-x-4 ni \left(3x^{2}-4x\right)+\left(3x-4\right) sifatida qaytadan yozish.

$$\left(3x^{2}-4x\right)+\left(3x-4\right)$$

3x^{2}-4x ichida x ni ajrating.

$$x\left(3x-4\right)+3x-4$$

Distributiv funktsiyasidan foydalangan holda 3x-4 umumiy terminini chiqaring.

$$\left(3x-4\right)\left(x+1\right)$$

Tenglamani yechish uchun 3x-4=0 va x+1=0 ni yeching.

$$x=\frac{4}{3} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

7 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(7x-7\right)\left(x+1\right)=\left(x+1\right)^{2}$$

7x-7 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$7x^{2}-7=\left(x+1\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$7x^{2}-7=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$7x^{2}-7-x^{2}=2x+1$$

6x^{2} ni olish uchun 7x^{2} va -x^{2} ni birlashtirish.

$$6x^{2}-7=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$6x^{2}-7-2x=1$$

Ikkala tarafdan 1 ni ayirish.

$$6x^{2}-7-2x-1=0$$

-8 olish uchun -7 dan 1 ni ayirish.

$$6x^{2}-8-2x=0$$

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.

$$6x^{2}-2x-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -2 ni b va -8 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 6\left(-8\right)}}{2\times 6}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 6\left(-8\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-24\left(-8\right)}}{2\times 6}$$

-24 ni -8 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2\times 6}$$

4 ni 192 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2\times 6}$$

196 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±14}{2\times 6}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±14}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{2±14}{12}$$

x=\frac{2±14}{12} tenglamasini yeching, bunda ± musbat. 2 ni 14 ga qo'shish.

$$x=\frac{16}{12}$$

\frac{16}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{4}{3}$$

x=\frac{2±14}{12} tenglamasini yeching, bunda ± manfiy. 2 dan 14 ni ayirish.

$$x=-\frac{12}{12}$$

-12 ni 12 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=\frac{4}{3} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

7 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(7x-7\right)\left(x+1\right)=\left(x+1\right)^{2}$$

7x-7 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$7x^{2}-7=\left(x+1\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$7x^{2}-7=x^{2}+2x+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$7x^{2}-7-x^{2}=2x+1$$

6x^{2} ni olish uchun 7x^{2} va -x^{2} ni birlashtirish.

$$6x^{2}-7=2x+1$$

Ikkala tarafdan 2x ni ayirish.

$$6x^{2}-7-2x=1$$

7 ni ikki tarafga qo’shing.

$$6x^{2}-2x=1+7$$

8 olish uchun 1 va 7'ni qo'shing.

$$6x^{2}-2x=8$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-2x}{6}=\frac{8}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{2}{6}\right)x=\frac{8}{6}$$

\frac{-2}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{3}x=\frac{8}{6}$$

\frac{8}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{3}x=\frac{4}{3}$$

-\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{6} olish uchun. Keyin, -\frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=\frac{4}{3}+\left(-\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{6} kvadratini chiqarish.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{4}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{4}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{49}{36}$$

x^{2}-\frac{1}{3}x+\frac{1}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{49}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Qisqartirish.

$$x-\frac{1}{6}=\frac{7}{6} x-\frac{1}{6}=-\frac{7}{6}$$

\frac{1}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4}{3} x=-1$$