(x-1)^3=x^2(x-1)-(x+3)(x-2)-19 eching

$$( x - 1 ) ^ { 3 } = x ^ { 2 } ( x - 1 ) - ( x + 3 ) ( x - 2 ) - 19$$

x uchun yechish

$$x=-2$$

$$x=6$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(x-1\right)^{3} kengaytirilishi uchun ishlating.

$$x^{3}-3x^{2}+3x-1=x^{2}\left(x-1\right)-\left(x+3\right)\left(x-2\right)-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x+3\right)\left(x-2\right)-19$$

x+3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x^{2}+x-6\right)-19$$

x^{2}+x-6 teskarisini topish uchun har birining teskarisini toping.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-x^{2}-x+6-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x+6-19$$

-13 olish uchun 6 dan 19 ni ayirish.

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x-13$$

Ikkala tarafdan x^{3} ni ayirish.

$$x^{3}-3x^{2}+3x-1-x^{3}=-2x^{2}-x-13$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$-3x^{2}+3x-1=-2x^{2}-x-13$$

2x^{2} ni ikki tarafga qo’shing.

$$-3x^{2}+3x-1+2x^{2}=-x-13$$

-x^{2} ni olish uchun -3x^{2} va 2x^{2} ni birlashtirish.

$$-x^{2}+3x-1=-x-13$$

x ni ikki tarafga qo’shing.

$$-x^{2}+3x-1+x=-13$$

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

$$-x^{2}+4x-1=-13$$

13 ni ikki tarafga qo’shing.

$$-x^{2}+4x-1+13=0$$

12 olish uchun -1 va 13'ni qo'shing.

$$-x^{2}+4x+12=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+12 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=4 ab=-12=-12$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga 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 – 4 yigʻindisini beruvchi juftlik.

$$a=6 b=-2$$

-x^{2}+4x+12 ni \left(-x^{2}+6x\right)+\left(-2x+12\right) sifatida qaytadan yozish.

$$\left(-x^{2}+6x\right)+\left(-2x+12\right)$$

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

$$-x\left(x-6\right)-2\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(-x-2\right)$$

Tenglamani yechish uchun x-6=0 va -x-2=0 ni yeching.

$$x=6 x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(x-1\right)^{3} kengaytirilishi uchun ishlating.

$$x^{3}-3x^{2}+3x-1=x^{2}\left(x-1\right)-\left(x+3\right)\left(x-2\right)-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x+3\right)\left(x-2\right)-19$$

x+3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x^{2}+x-6\right)-19$$

x^{2}+x-6 teskarisini topish uchun har birining teskarisini toping.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-x^{2}-x+6-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x+6-19$$

-13 olish uchun 6 dan 19 ni ayirish.

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x-13$$

Ikkala tarafdan x^{3} ni ayirish.

$$x^{3}-3x^{2}+3x-1-x^{3}=-2x^{2}-x-13$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$-3x^{2}+3x-1=-2x^{2}-x-13$$

2x^{2} ni ikki tarafga qo’shing.

$$-3x^{2}+3x-1+2x^{2}=-x-13$$

-x^{2} ni olish uchun -3x^{2} va 2x^{2} ni birlashtirish.

$$-x^{2}+3x-1=-x-13$$

x ni ikki tarafga qo’shing.

$$-x^{2}+3x-1+x=-13$$

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

$$-x^{2}+4x-1=-13$$

13 ni ikki tarafga qo’shing.

$$-x^{2}+4x-1+13=0$$

12 olish uchun -1 va 13'ni qo'shing.

$$-x^{2}+4x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 4 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+4\times 12}}{2\left(-1\right)}$$

4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+48}}{2\left(-1\right)}$$

16 ni 48 ga qo'shish.

$$x=\frac{-4±\sqrt{64}}{2\left(-1\right)}$$

64 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±8}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-4±8}{-2}$$

x=\frac{-4±8}{-2} tenglamasini yeching, bunda ± musbat. -4 ni 8 ga qo'shish.

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

4 ni -2 ga bo'lish.

$$x=-2$$

x=\frac{-4±8}{-2} tenglamasini yeching, bunda ± manfiy. -4 dan 8 ni ayirish.

$$x=-\frac{12}{-2}$$

-12 ni -2 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=-2 x=6$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(x-1\right)^{3} kengaytirilishi uchun ishlating.

$$x^{3}-3x^{2}+3x-1=x^{2}\left(x-1\right)-\left(x+3\right)\left(x-2\right)-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x+3\right)\left(x-2\right)-19$$

x+3 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-\left(x^{2}+x-6\right)-19$$

x^{2}+x-6 teskarisini topish uchun har birining teskarisini toping.

$$x^{3}-3x^{2}+3x-1=x^{3}-x^{2}-x^{2}-x+6-19$$

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

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x+6-19$$

-13 olish uchun 6 dan 19 ni ayirish.

$$x^{3}-3x^{2}+3x-1=x^{3}-2x^{2}-x-13$$

Ikkala tarafdan x^{3} ni ayirish.

$$x^{3}-3x^{2}+3x-1-x^{3}=-2x^{2}-x-13$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$-3x^{2}+3x-1=-2x^{2}-x-13$$

2x^{2} ni ikki tarafga qo’shing.

$$-3x^{2}+3x-1+2x^{2}=-x-13$$

-x^{2} ni olish uchun -3x^{2} va 2x^{2} ni birlashtirish.

$$-x^{2}+3x-1=-x-13$$

x ni ikki tarafga qo’shing.

$$-x^{2}+3x-1+x=-13$$

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

$$-x^{2}+4x-1=-13$$

1 ni ikki tarafga qo’shing.

$$-x^{2}+4x=-13+1$$

-12 olish uchun -13 va 1'ni qo'shing.

$$-x^{2}+4x=-12$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+4x}{-1}=-\frac{12}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{4}{-1}x=-\frac{12}{-1}$$

4 ni -1 ga bo'lish.

$$x^{2}-4x=-\frac{12}{-1}$$

-12 ni -1 ga bo'lish.

$$x^{2}-4x=12$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=12+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=12+4$$

12 ni 4 ga qo'shish.

$$x^{2}-4x+4=16$$

x^{2}-4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-2\right)^{2}=16$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{16}$$

Qisqartirish.

$$x-2=4 x-2=-4$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=-2$$