x^2=11x+12 eching

$$x ^ { 2 } = 11 x + 12$$

x uchun yechish

$$x=-1$$

$$x=12$$

Faktorlash qadamlari

Ikkala tarafdan 11x ni ayirish.

$$x^{2}-11x=12$$

Ikkala tarafdan 12 ni ayirish.

$$x^{2}-11x-12=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-11x-12 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-11 ab=-12$$

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 – -11 yigʻindisini beruvchi juftlik.

$$a=-12 b=1$$

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

$$\left(x-12\right)\left(x+1\right)$$

Tenglamani yechish uchun x-12=0 va x+1=0 ni yeching.

$$x=12 x=-1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 11x ni ayirish.

$$x^{2}-11x=12$$

Ikkala tarafdan 12 ni ayirish.

$$x^{2}-11x-12=0$$

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.

$$a+b=-11 ab=1\left(-12\right)=-12$$

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 – -11 yigʻindisini beruvchi juftlik.

$$a=-12 b=1$$

x^{2}-11x-12 ni \left(x^{2}-12x\right)+\left(x-12\right) sifatida qaytadan yozish.

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

x^{2}-12x ichida x ni ajrating.

$$x\left(x-12\right)+x-12$$

Distributiv funktsiyasidan foydalangan holda x-12 umumiy terminini chiqaring.

$$\left(x-12\right)\left(x+1\right)$$

Tenglamani yechish uchun x-12=0 va x+1=0 ni yeching.

$$x=12 x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 11x ni ayirish.

$$x^{2}-11x=12$$

Ikkala tarafdan 12 ni ayirish.

$$x^{2}-11x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -11 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-12\right)}}{2}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-12\right)}}{2}$$

-4 ni -12 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+48}}{2}$$

121 ni 48 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{169}}{2}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-11\right)±13}{2}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±13}{2}$$

x=\frac{11±13}{2} tenglamasini yeching, bunda ± musbat. 11 ni 13 ga qo'shish.

$$x=\frac{24}{2}$$

24 ni 2 ga bo'lish.

$$x=12$$

x=\frac{11±13}{2} tenglamasini yeching, bunda ± manfiy. 11 dan 13 ni ayirish.

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

-2 ni 2 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=12 x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 11x ni ayirish.

$$x^{2}-11x=12$$

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

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=12+\left(-\frac{11}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{2} kvadratini chiqarish.

$$x^{2}-11x+\frac{121}{4}=12+\frac{121}{4}$$

12 ni \frac{121}{4} ga qo'shish.

$$x^{2}-11x+\frac{121}{4}=\frac{169}{4}$$

x^{2}-11x+\frac{121}{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.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{169}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Qisqartirish.

$$x-\frac{11}{2}=\frac{13}{2} x-\frac{11}{2}=-\frac{13}{2}$$

\frac{11}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=12 x=-1$$