x^2-3x^2-2x+12=0 eching

$$x ^ { 2 } - 3 x ^ { 2 } - 2 x + 12 = 0$$

x uchun yechish

$$x=-3$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

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

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

Ikki tarafini 2 ga bo‘ling.

$$-x^{2}-x+6=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=-6=-6$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -6-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-6 2,-3$$

Har bir juftlik yigʻindisini hisoblang.

$$1-6=-5 2-3=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=2 b=-3$$

-x^{2}-x+6 ni \left(-x^{2}+2x\right)+\left(-3x+6\right) sifatida qaytadan yozish.

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

Birinchi guruhda x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$x\left(-x+2\right)+3\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(x+3\right)$$

Tenglamani yechish uchun -x+2=0 va x+3=0 ni yeching.

$$x=2 x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

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

-2 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+8\times 12}}{2\left(-2\right)}$$

8 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+96}}{2\left(-2\right)}$$

4 ni 96 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{100}}{2\left(-2\right)}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±10}{2\left(-2\right)}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±10}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{2±10}{-4}$$

x=\frac{2±10}{-4} tenglamasini yeching, bunda ± musbat. 2 ni 10 ga qo'shish.

$$x=\frac{12}{-4}$$

12 ni -4 ga bo'lish.

$$x=-3$$

x=\frac{2±10}{-4} tenglamasini yeching, bunda ± manfiy. 2 dan 10 ni ayirish.

$$x=-\frac{8}{-4}$$

-8 ni -4 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=-3 x=2$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

Ikkala tarafdan 12 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

Ikki tarafini -2 ga bo‘ling.

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

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

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

-2 ni -2 ga bo'lish.

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

-12 ni -2 ga bo'lish.

$$x^{2}+x=6$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=6+\frac{1}{4}$$

6 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{25}{4}$$

x^{2}+x+\frac{1}{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.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{5}{2} x+\frac{1}{2}=-\frac{5}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=2 x=-3$$