(x-1)^2+(x+2)^2-(x-3)(x+3)=22 eching

$$( x - 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } - ( x - 3 ) ( x + 3 ) = 22$$

x uchun yechish

$$x=-4$$

$$x=2$$

Faktorlash qadamlari

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

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

2x ni olish uchun -2x va 4x ni birlashtirish.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

5 olish uchun 1 va 4'ni qo'shing.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Hisoblang: \left(x-3\right)\left(x+3\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 3 kvadratini chiqarish.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

x^{2}-9 teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}+2x+5-x^{2}+9=22$$

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

$$x^{2}+2x+5+9=22$$

14 olish uchun 5 va 9'ni qo'shing.

$$x^{2}+2x+14=22$$

Ikkala tarafdan 22 ni ayirish.

$$x^{2}+2x+14-22=0$$

-8 olish uchun 14 dan 22 ni ayirish.

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

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}+2x-8 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=2 ab=-8$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -8-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,8 -2,4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+8=7 -2+4=2$$

Yechim – 2 yigʻindisini beruvchi juftlik.

$$a=-2 b=4$$

Faktorlangan \left(x+a\right)\left(x+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(x-2\right)\left(x+4\right)$$

Tenglamani yechish uchun x-2=0 va x+4=0 ni yeching.

$$x=2 x=-4$$

Guruhlash orqali faktorlash qadamlari

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

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

2x ni olish uchun -2x va 4x ni birlashtirish.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

5 olish uchun 1 va 4'ni qo'shing.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Hisoblang: \left(x-3\right)\left(x+3\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 3 kvadratini chiqarish.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

x^{2}-9 teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}+2x+5-x^{2}+9=22$$

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

$$x^{2}+2x+5+9=22$$

14 olish uchun 5 va 9'ni qo'shing.

$$x^{2}+2x+14=22$$

Ikkala tarafdan 22 ni ayirish.

$$x^{2}+2x+14-22=0$$

-8 olish uchun 14 dan 22 ni ayirish.

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

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=2 ab=1\left(-8\right)=-8$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -8-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,8 -2,4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+8=7 -2+4=2$$

Yechim – 2 yigʻindisini beruvchi juftlik.

$$a=-2 b=4$$

x^{2}+2x-8 ni \left(x^{2}-2x\right)+\left(4x-8\right) sifatida qaytadan yozish.

$$\left(x^{2}-2x\right)+\left(4x-8\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 4 ni faktordan chiqaring.

$$x\left(x-2\right)+4\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(x+4\right)$$

Tenglamani yechish uchun x-2=0 va x+4=0 ni yeching.

$$x=2 x=-4$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

2x ni olish uchun -2x va 4x ni birlashtirish.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

5 olish uchun 1 va 4'ni qo'shing.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Hisoblang: \left(x-3\right)\left(x+3\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 3 kvadratini chiqarish.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

x^{2}-9 teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}+2x+5-x^{2}+9=22$$

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

$$x^{2}+2x+5+9=22$$

14 olish uchun 5 va 9'ni qo'shing.

$$x^{2}+2x+14=22$$

Ikkala tarafdan 22 ni ayirish.

$$x^{2}+2x+14-22=0$$

-8 olish uchun 14 dan 22 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 2 ni b va -8 ni c bilan almashtiring.

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

2 kvadratini chiqarish.

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

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+32}}{2}$$

4 ni 32 ga qo'shish.

$$x=\frac{-2±\sqrt{36}}{2}$$

36 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±6}{2}$$

x=\frac{-2±6}{2} tenglamasini yeching, bunda ± musbat. -2 ni 6 ga qo'shish.

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

4 ni 2 ga bo'lish.

$$x=2$$

x=\frac{-2±6}{2} tenglamasini yeching, bunda ± manfiy. -2 dan 6 ni ayirish.

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

-8 ni 2 ga bo'lish.

$$x=-4$$

Tenglama yechildi.

$$x=2 x=-4$$

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.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)=22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)=22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)=22$$

2x ni olish uchun -2x va 4x ni birlashtirish.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)=22$$

5 olish uchun 1 va 4'ni qo'shing.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)=22$$

Hisoblang: \left(x-3\right)\left(x+3\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 3 kvadratini chiqarish.

$$2x^{2}+2x+5-\left(x^{2}-9\right)=22$$

x^{2}-9 teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}+2x+5-x^{2}+9=22$$

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

$$x^{2}+2x+5+9=22$$

14 olish uchun 5 va 9'ni qo'shing.

$$x^{2}+2x+14=22$$

Ikkala tarafdan 14 ni ayirish.

$$x^{2}+2x=22-14$$

8 olish uchun 22 dan 14 ni ayirish.

$$x^{2}+2x=8$$

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

$$x^{2}+2x+1^{2}=8+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=8+1$$

8 ni 1 ga qo'shish.

$$x^{2}+2x+1=9$$

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

$$\left(x+1\right)^{2}=9$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Qisqartirish.

$$x+1=3 x+1=-3$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=2 x=-4$$