4x^2-4x+1=x^2-6x+9 eching

$$4 x ^ { 2 } - 4 x + 1 = x ^ { 2 } - 6 x + 9$$

x uchun yechish

$$x=-2$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1,333333333$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-4x+1-x^{2}=-6x+9$$

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

$$3x^{2}-4x+1=-6x+9$$

6x ni ikki tarafga qo’shing.

$$3x^{2}-4x+1+6x=9$$

2x ni olish uchun -4x va 6x ni birlashtirish.

$$3x^{2}+2x+1=9$$

Ikkala tarafdan 9 ni ayirish.

$$3x^{2}+2x+1-9=0$$

-8 olish uchun 1 dan 9 ni ayirish.

$$3x^{2}+2x-8=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=2 ab=3\left(-8\right)=-24$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -24-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,24 -2,12 -3,8 -4,6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Yechim – 2 yigʻindisini beruvchi juftlik.

$$a=-4 b=6$$

3x^{2}+2x-8 ni \left(3x^{2}-4x\right)+\left(6x-8\right) sifatida qaytadan yozish.

$$\left(3x^{2}-4x\right)+\left(6x-8\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$x\left(3x-4\right)+2\left(3x-4\right)$$

Distributiv funktsiyasidan foydalangan holda 3x-4 umumiy terminini chiqaring.

$$\left(3x-4\right)\left(x+2\right)$$

Tenglamani yechish uchun 3x-4=0 va x+2=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-4x+1-x^{2}=-6x+9$$

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

$$3x^{2}-4x+1=-6x+9$$

6x ni ikki tarafga qo’shing.

$$3x^{2}-4x+1+6x=9$$

2x ni olish uchun -4x va 6x ni birlashtirish.

$$3x^{2}+2x+1=9$$

Ikkala tarafdan 9 ni ayirish.

$$3x^{2}+2x+1-9=0$$

-8 olish uchun 1 dan 9 ni ayirish.

$$3x^{2}+2x-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 2 ni b va -8 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\left(-8\right)}}{2\times 3}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 3\left(-8\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-12\left(-8\right)}}{2\times 3}$$

-12 ni -8 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+96}}{2\times 3}$$

4 ni 96 ga qo'shish.

$$x=\frac{-2±\sqrt{100}}{2\times 3}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±10}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-2±10}{6}$$

x=\frac{-2±10}{6} tenglamasini yeching, bunda ± musbat. -2 ni 10 ga qo'shish.

$$x=\frac{8}{6}$$

\frac{8}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{4}{3}$$

x=\frac{-2±10}{6} tenglamasini yeching, bunda ± manfiy. -2 dan 10 ni ayirish.

$$x=-\frac{12}{6}$$

-12 ni 6 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-4x+1-x^{2}=-6x+9$$

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

$$3x^{2}-4x+1=-6x+9$$

6x ni ikki tarafga qo’shing.

$$3x^{2}-4x+1+6x=9$$

2x ni olish uchun -4x va 6x ni birlashtirish.

$$3x^{2}+2x+1=9$$

Ikkala tarafdan 1 ni ayirish.

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

8 olish uchun 9 dan 1 ni ayirish.

$$3x^{2}+2x=8$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+2x}{3}=\frac{8}{3}$$

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

$$x^{2}+\frac{2}{3}x=\frac{8}{3}$$

\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{3} olish uchun. Keyin, \frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\frac{8}{3}+\left(\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{3} kvadratini chiqarish.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{8}{3}+\frac{1}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{8}{3} ni \frac{1}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{25}{9}$$

x^{2}+\frac{2}{3}x+\frac{1}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{25}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Qisqartirish.

$$x+\frac{1}{3}=\frac{5}{3} x+\frac{1}{3}=-\frac{5}{3}$$

Tenglamaning ikkala tarafidan \frac{1}{3} ni ayirish.

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