3(x^2-4)=(x+8)(x-2) eching

$$3 ( x ^ { 2 } - 4 ) = ( x + 8 ) ( x - 2 )$$

x uchun yechish

$$x=1$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

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

$$3x^{2}-12=\left(x+8\right)\left(x-2\right)$$

x+8 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-12=x^{2}+6x-16$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x^{2}-12-x^{2}=6x-16$$

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

$$2x^{2}-12=6x-16$$

Ikkala tarafdan 6x ni ayirish.

$$2x^{2}-12-6x=-16$$

16 ni ikki tarafga qo’shing.

$$2x^{2}-12-6x+16=0$$

4 olish uchun -12 va 16'ni qo'shing.

$$2x^{2}+4-6x=0$$

Ikki tarafini 2 ga bo‘ling.

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

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

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

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

$$a+b=-3 ab=1\times 2=2$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. Faqat bundan juftlik tizim yechimidir.

$$a=-2 b=-1$$

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

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

Birinchi guruhda x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

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

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

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

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

$$x=2 x=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$3x^{2}-12=\left(x+8\right)\left(x-2\right)$$

x+8 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-12=x^{2}+6x-16$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x^{2}-12-x^{2}=6x-16$$

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

$$2x^{2}-12=6x-16$$

Ikkala tarafdan 6x ni ayirish.

$$2x^{2}-12-6x=-16$$

16 ni ikki tarafga qo’shing.

$$2x^{2}-12-6x+16=0$$

4 olish uchun -12 va 16'ni qo'shing.

$$2x^{2}+4-6x=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$2x^{2}-6x+4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -6 ni b va 4 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\times 4}}{2\times 2}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\times 4}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\times 4}}{2\times 2}$$

-8 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-32}}{2\times 2}$$

36 ni -32 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{4}}{2\times 2}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±2}{2\times 2}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±2}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{6±2}{4}$$

x=\frac{6±2}{4} tenglamasini yeching, bunda ± musbat. 6 ni 2 ga qo'shish.

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

8 ni 4 ga bo'lish.

$$x=2$$

x=\frac{6±2}{4} tenglamasini yeching, bunda ± manfiy. 6 dan 2 ni ayirish.

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

4 ni 4 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=2 x=1$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$3x^{2}-12=\left(x+8\right)\left(x-2\right)$$

x+8 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-12=x^{2}+6x-16$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x^{2}-12-x^{2}=6x-16$$

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

$$2x^{2}-12=6x-16$$

Ikkala tarafdan 6x ni ayirish.

$$2x^{2}-12-6x=-16$$

12 ni ikki tarafga qo’shing.

$$2x^{2}-6x=-16+12$$

-4 olish uchun -16 va 12'ni qo'shing.

$$2x^{2}-6x=-4$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-6x}{2}=-\frac{4}{2}$$

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

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

-6 ni 2 ga bo'lish.

$$x^{2}-3x=-\frac{4}{2}$$

-4 ni 2 ga bo'lish.

$$x^{2}-3x=-2$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=-2+\frac{9}{4}$$

-2 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{1}{4}$$

x^{2}-3x+\frac{9}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-\frac{3}{2}=\frac{1}{2} x-\frac{3}{2}=-\frac{1}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=1$$