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

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

x uchun yechish

$$x=\frac{1}{2}=0,5$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

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

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

2 ga x^{2}-2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-4x+2=1-x$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}-4x+2-1=-x$$

1 olish uchun 2 dan 1 ni ayirish.

$$2x^{2}-4x+1=-x$$

x ni ikki tarafga qo’shing.

$$2x^{2}-4x+1+x=0$$

-3x ni olish uchun -4x va x ni birlashtirish.

$$2x^{2}-3x+1=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx+1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=2\times 1=2$$

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.

$$a=-2 b=-1$$

2x^{2}-3x+1 ni \left(2x^{2}-2x\right)+\left(-x+1\right) sifatida qaytadan yozish.

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

Birinchi guruhda 2x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$2x\left(x-1\right)-\left(x-1\right)$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(2x-1\right)$$

Tenglamani yechish uchun x-1=0 va 2x-1=0 ni yeching.

$$x=1 x=\frac{1}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

2 ga x^{2}-2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-4x+2=1-x$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}-4x+2-1=-x$$

1 olish uchun 2 dan 1 ni ayirish.

$$2x^{2}-4x+1=-x$$

x ni ikki tarafga qo’shing.

$$2x^{2}-4x+1+x=0$$

-3x ni olish uchun -4x va x ni birlashtirish.

$$2x^{2}-3x+1=0$$

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 1 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8}}{2\times 2}$$

9 ni -8 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{1}}{2\times 2}$$

1 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±1}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±1}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±1}{4}$$

x=\frac{3±1}{4} tenglamasini yeching, bunda ± musbat. 3 ni 1 ga qo'shish.

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

4 ni 4 ga bo'lish.

$$x=1$$

x=\frac{3±1}{4} tenglamasini yeching, bunda ± manfiy. 3 dan 1 ni ayirish.

$$x=\frac{2}{4}$$

\frac{2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{2}$$

Tenglama yechildi.

$$x=1 x=\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

2 ga x^{2}-2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-4x+2=1-x$$

x ni ikki tarafga qo’shing.

$$2x^{2}-4x+2+x=1$$

-3x ni olish uchun -4x va x ni birlashtirish.

$$2x^{2}-3x+2=1$$

Ikkala tarafdan 2 ni ayirish.

$$2x^{2}-3x=1-2$$

-1 olish uchun 1 dan 2 ni ayirish.

$$2x^{2}-3x=-1$$

Ikki tarafini 2 ga bo‘ling.

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

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-\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.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}$$

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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{16}$$

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.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{1}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Qisqartirish.

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

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=1 x=\frac{1}{2}$$