4left(x-3right)^2=x(x-3) eching

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

x uchun yechish

$$x=3$$

$$x=4$$

Guruhlash orqali faktorlash qadamlari

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

$$4\left(x^{2}-6x+9\right)=x\left(x-3\right)$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x\left(x-3\right)$$

x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x^{2}-3x$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-24x+36-x^{2}=-3x$$

3x^{2} ni olish uchun 4x^{2} va -x^{2} ni birlashtirish.

$$3x^{2}-24x+36=-3x$$

3x ni ikki tarafga qo’shing.

$$3x^{2}-24x+36+3x=0$$

-21x ni olish uchun -24x va 3x ni birlashtirish.

$$3x^{2}-21x+36=0$$

Ikki tarafini 3 ga bo‘ling.

$$x^{2}-7x+12=0$$

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.

$$a+b=-7 ab=1\times 12=12$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-12 -2,-6 -3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Yechim – -7 yigʻindisini beruvchi juftlik.

$$a=-4 b=-3$$

x^{2}-7x+12 ni \left(x^{2}-4x\right)+\left(-3x+12\right) sifatida qaytadan yozish.

$$\left(x^{2}-4x\right)+\left(-3x+12\right)$$

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

$$x\left(x-4\right)-3\left(x-4\right)$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(x-3\right)$$

Tenglamani yechish uchun x-4=0 va x-3=0 ni yeching.

$$x=4 x=3$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$4\left(x^{2}-6x+9\right)=x\left(x-3\right)$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x\left(x-3\right)$$

x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x^{2}-3x$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-24x+36-x^{2}=-3x$$

3x^{2} ni olish uchun 4x^{2} va -x^{2} ni birlashtirish.

$$3x^{2}-24x+36=-3x$$

3x ni ikki tarafga qo’shing.

$$3x^{2}-24x+36+3x=0$$

-21x ni olish uchun -24x va 3x ni birlashtirish.

$$3x^{2}-21x+36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -21 ni b va 36 ni c bilan almashtiring.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 3\times 36}}{2\times 3}$$

-21 kvadratini chiqarish.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 3\times 36}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-21\right)±\sqrt{441-12\times 36}}{2\times 3}$$

-12 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-21\right)±\sqrt{441-432}}{2\times 3}$$

441 ni -432 ga qo'shish.

$$x=\frac{-\left(-21\right)±\sqrt{9}}{2\times 3}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-21\right)±3}{2\times 3}$$

-21 ning teskarisi 21 ga teng.

$$x=\frac{21±3}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{21±3}{6}$$

x=\frac{21±3}{6} tenglamasini yeching, bunda ± musbat. 21 ni 3 ga qo'shish.

$$x=\frac{24}{6}$$

24 ni 6 ga bo'lish.

$$x=4$$

x=\frac{21±3}{6} tenglamasini yeching, bunda ± manfiy. 21 dan 3 ni ayirish.

$$x=\frac{18}{6}$$

18 ni 6 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=4 x=3$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$4\left(x^{2}-6x+9\right)=x\left(x-3\right)$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x\left(x-3\right)$$

x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x^{2}-3x$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-24x+36-x^{2}=-3x$$

3x^{2} ni olish uchun 4x^{2} va -x^{2} ni birlashtirish.

$$3x^{2}-24x+36=-3x$$

3x ni ikki tarafga qo’shing.

$$3x^{2}-24x+36+3x=0$$

-21x ni olish uchun -24x va 3x ni birlashtirish.

$$3x^{2}-21x+36=0$$

Ikkala tarafdan 36 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3x^{2}-21x=-36$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-21x}{3}=-\frac{36}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{21}{3}\right)x=-\frac{36}{3}$$

-21 ni 3 ga bo'lish.

$$x^{2}-7x=-\frac{36}{3}$$

-36 ni 3 ga bo'lish.

$$x^{2}-7x=-12$$

-7 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{2} olish uchun. Keyin, -\frac{7}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-12+\left(-\frac{7}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=-12+\frac{49}{4}$$

-12 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{1}{4}$$

x^{2}-7x+\frac{49}{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.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{1}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{1}{2} x-\frac{7}{2}=-\frac{1}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=3$$