(2x-1)^2/3-(x-2)(1-2x)/6=(1-2x)^2 eching

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

x uchun yechish

$$x=\frac{2}{7}\approx 0,285714286$$

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

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 6 ga, 3,6 ning eng kichik karralisiga ko‘paytiring.

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

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

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

2 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-\left(x-2\right)\left(1-2x\right)=6\left(1-2x\right)^{2}$$

x-2 ga 1-2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-8x+2-\left(5x-2x^{2}-2\right)=6\left(1-2x\right)^{2}$$

5x-2x^{2}-2 teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-8x+2-5x+2x^{2}+2=6\left(1-2x\right)^{2}$$

-13x ni olish uchun -8x va -5x ni birlashtirish.

$$8x^{2}-13x+2+2x^{2}+2=6\left(1-2x\right)^{2}$$

10x^{2} ni olish uchun 8x^{2} va 2x^{2} ni birlashtirish.

$$10x^{2}-13x+2+2=6\left(1-2x\right)^{2}$$

4 olish uchun 2 va 2'ni qo'shing.

$$10x^{2}-13x+4=6\left(1-2x\right)^{2}$$

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

$$10x^{2}-13x+4=6\left(1-4x+4x^{2}\right)$$

6 ga 1-4x+4x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x^{2}-13x+4=6-24x+24x^{2}$$

Ikkala tarafdan 6 ni ayirish.

$$10x^{2}-13x+4-6=-24x+24x^{2}$$

-2 olish uchun 4 dan 6 ni ayirish.

$$10x^{2}-13x-2=-24x+24x^{2}$$

24x ni ikki tarafga qo’shing.

$$10x^{2}-13x-2+24x=24x^{2}$$

11x ni olish uchun -13x va 24x ni birlashtirish.

$$10x^{2}+11x-2=24x^{2}$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$10x^{2}+11x-2-24x^{2}=0$$

-14x^{2} ni olish uchun 10x^{2} va -24x^{2} ni birlashtirish.

$$-14x^{2}+11x-2=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -14x^{2}+ax+bx-2 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=11 ab=-14\left(-2\right)=28$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 28-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,28 2,14 4,7$$

Har bir juftlik yigʻindisini hisoblang.

$$1+28=29 2+14=16 4+7=11$$

Yechim – 11 yigʻindisini beruvchi juftlik.

$$a=7 b=4$$

-14x^{2}+11x-2 ni \left(-14x^{2}+7x\right)+\left(4x-2\right) sifatida qaytadan yozish.

$$\left(-14x^{2}+7x\right)+\left(4x-2\right)$$

Birinchi guruhda -7x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$-7x\left(2x-1\right)+2\left(2x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-1 umumiy terminini chiqaring.

$$\left(2x-1\right)\left(-7x+2\right)$$

Tenglamani yechish uchun 2x-1=0 va -7x+2=0 ni yeching.

$$x=\frac{1}{2} x=\frac{2}{7}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 6 ga, 3,6 ning eng kichik karralisiga ko‘paytiring.

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

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

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

2 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-\left(x-2\right)\left(1-2x\right)=6\left(1-2x\right)^{2}$$

x-2 ga 1-2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-8x+2-\left(5x-2x^{2}-2\right)=6\left(1-2x\right)^{2}$$

5x-2x^{2}-2 teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-8x+2-5x+2x^{2}+2=6\left(1-2x\right)^{2}$$

-13x ni olish uchun -8x va -5x ni birlashtirish.

$$8x^{2}-13x+2+2x^{2}+2=6\left(1-2x\right)^{2}$$

10x^{2} ni olish uchun 8x^{2} va 2x^{2} ni birlashtirish.

$$10x^{2}-13x+2+2=6\left(1-2x\right)^{2}$$

4 olish uchun 2 va 2'ni qo'shing.

$$10x^{2}-13x+4=6\left(1-2x\right)^{2}$$

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

$$10x^{2}-13x+4=6\left(1-4x+4x^{2}\right)$$

6 ga 1-4x+4x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x^{2}-13x+4=6-24x+24x^{2}$$

Ikkala tarafdan 6 ni ayirish.

$$10x^{2}-13x+4-6=-24x+24x^{2}$$

-2 olish uchun 4 dan 6 ni ayirish.

$$10x^{2}-13x-2=-24x+24x^{2}$$

24x ni ikki tarafga qo’shing.

$$10x^{2}-13x-2+24x=24x^{2}$$

11x ni olish uchun -13x va 24x ni birlashtirish.

$$10x^{2}+11x-2=24x^{2}$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$10x^{2}+11x-2-24x^{2}=0$$

-14x^{2} ni olish uchun 10x^{2} va -24x^{2} ni birlashtirish.

$$-14x^{2}+11x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -14 ni a, 11 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-14\right)\left(-2\right)}}{2\left(-14\right)}$$

11 kvadratini chiqarish.

$$x=\frac{-11±\sqrt{121-4\left(-14\right)\left(-2\right)}}{2\left(-14\right)}$$

-4 ni -14 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121+56\left(-2\right)}}{2\left(-14\right)}$$

56 ni -2 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121-112}}{2\left(-14\right)}$$

121 ni -112 ga qo'shish.

$$x=\frac{-11±\sqrt{9}}{2\left(-14\right)}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-11±3}{2\left(-14\right)}$$

2 ni -14 marotabaga ko'paytirish.

$$x=\frac{-11±3}{-28}$$

x=\frac{-11±3}{-28} tenglamasini yeching, bunda ± musbat. -11 ni 3 ga qo'shish.

$$x=-\frac{8}{-28}$$

\frac{-8}{-28} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{2}{7}$$

x=\frac{-11±3}{-28} tenglamasini yeching, bunda ± manfiy. -11 dan 3 ni ayirish.

$$x=-\frac{14}{-28}$$

\frac{-14}{-28} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=\frac{2}{7} x=\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 6 ga, 3,6 ning eng kichik karralisiga ko‘paytiring.

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

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

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

2 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-\left(x-2\right)\left(1-2x\right)=6\left(1-2x\right)^{2}$$

x-2 ga 1-2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-8x+2-\left(5x-2x^{2}-2\right)=6\left(1-2x\right)^{2}$$

5x-2x^{2}-2 teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-8x+2-5x+2x^{2}+2=6\left(1-2x\right)^{2}$$

-13x ni olish uchun -8x va -5x ni birlashtirish.

$$8x^{2}-13x+2+2x^{2}+2=6\left(1-2x\right)^{2}$$

10x^{2} ni olish uchun 8x^{2} va 2x^{2} ni birlashtirish.

$$10x^{2}-13x+2+2=6\left(1-2x\right)^{2}$$

4 olish uchun 2 va 2'ni qo'shing.

$$10x^{2}-13x+4=6\left(1-2x\right)^{2}$$

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

$$10x^{2}-13x+4=6\left(1-4x+4x^{2}\right)$$

6 ga 1-4x+4x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x^{2}-13x+4=6-24x+24x^{2}$$

24x ni ikki tarafga qo’shing.

$$10x^{2}-13x+4+24x=6+24x^{2}$$

11x ni olish uchun -13x va 24x ni birlashtirish.

$$10x^{2}+11x+4=6+24x^{2}$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$10x^{2}+11x+4-24x^{2}=6$$

-14x^{2} ni olish uchun 10x^{2} va -24x^{2} ni birlashtirish.

$$-14x^{2}+11x+4=6$$

Ikkala tarafdan 4 ni ayirish.

$$-14x^{2}+11x=6-4$$

2 olish uchun 6 dan 4 ni ayirish.

$$-14x^{2}+11x=2$$

Ikki tarafini -14 ga bo‘ling.

$$\frac{-14x^{2}+11x}{-14}=\frac{2}{-14}$$

-14 ga bo'lish -14 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{11}{-14}x=\frac{2}{-14}$$

11 ni -14 ga bo'lish.

$$x^{2}-\frac{11}{14}x=\frac{2}{-14}$$

\frac{2}{-14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{11}{14}x=-\frac{1}{7}$$

-\frac{11}{14} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{28} olish uchun. Keyin, -\frac{11}{28} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{14}x+\left(-\frac{11}{28}\right)^{2}=-\frac{1}{7}+\left(-\frac{11}{28}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{28} kvadratini chiqarish.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=-\frac{1}{7}+\frac{121}{784}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{7} ni \frac{121}{784} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{9}{784}$$

x^{2}-\frac{11}{14}x+\frac{121}{784} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{28}\right)^{2}=\frac{9}{784}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{28}\right)^{2}}=\sqrt{\frac{9}{784}}$$

Qisqartirish.

$$x-\frac{11}{28}=\frac{3}{28} x-\frac{11}{28}=-\frac{3}{28}$$

\frac{11}{28} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1}{2} x=\frac{2}{7}$$