(2x-3)^2=(x-1)(x-4)+9x eching

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

x uchun yechish

$$x=\frac{1}{3}\approx 0,333333333$$

$$x=5$$

Guruhlash orqali faktorlash qadamlari

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

$$4x^{2}-12x+9=\left(x-1\right)\left(x-4\right)+9x$$

x-1 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-12x+9=x^{2}-5x+4+9x$$

4x ni olish uchun -5x va 9x ni birlashtirish.

$$4x^{2}-12x+9=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-12x+9-x^{2}=4x+4$$

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

$$3x^{2}-12x+9=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}-12x+9-4x=4$$

-16x ni olish uchun -12x va -4x ni birlashtirish.

$$3x^{2}-16x+9=4$$

Ikkala tarafdan 4 ni ayirish.

$$3x^{2}-16x+9-4=0$$

5 olish uchun 9 dan 4 ni ayirish.

$$3x^{2}-16x+5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx+5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-16 ab=3\times 5=15$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 15-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-15 -3,-5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-15=-16 -3-5=-8$$

Yechim – -16 yigʻindisini beruvchi juftlik.

$$a=-15 b=-1$$

3x^{2}-16x+5 ni \left(3x^{2}-15x\right)+\left(-x+5\right) sifatida qaytadan yozish.

$$\left(3x^{2}-15x\right)+\left(-x+5\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$3x\left(x-5\right)-\left(x-5\right)$$

Distributiv funktsiyasidan foydalangan holda x-5 umumiy terminini chiqaring.

$$\left(x-5\right)\left(3x-1\right)$$

Tenglamani yechish uchun x-5=0 va 3x-1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$4x^{2}-12x+9=\left(x-1\right)\left(x-4\right)+9x$$

x-1 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-12x+9=x^{2}-5x+4+9x$$

4x ni olish uchun -5x va 9x ni birlashtirish.

$$4x^{2}-12x+9=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-12x+9-x^{2}=4x+4$$

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

$$3x^{2}-12x+9=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}-12x+9-4x=4$$

-16x ni olish uchun -12x va -4x ni birlashtirish.

$$3x^{2}-16x+9=4$$

Ikkala tarafdan 4 ni ayirish.

$$3x^{2}-16x+9-4=0$$

5 olish uchun 9 dan 4 ni ayirish.

$$3x^{2}-16x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -16 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 3\times 5}}{2\times 3}$$

-16 kvadratini chiqarish.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 3\times 5}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-16\right)±\sqrt{256-12\times 5}}{2\times 3}$$

-12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-16\right)±\sqrt{256-60}}{2\times 3}$$

256 ni -60 ga qo'shish.

$$x=\frac{-\left(-16\right)±\sqrt{196}}{2\times 3}$$

196 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-16\right)±14}{2\times 3}$$

-16 ning teskarisi 16 ga teng.

$$x=\frac{16±14}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{16±14}{6}$$

x=\frac{16±14}{6} tenglamasini yeching, bunda ± musbat. 16 ni 14 ga qo'shish.

$$x=\frac{30}{6}$$

30 ni 6 ga bo'lish.

$$x=5$$

x=\frac{16±14}{6} tenglamasini yeching, bunda ± manfiy. 16 dan 14 ni ayirish.

$$x=\frac{2}{6}$$

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

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

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

$$4x^{2}-12x+9=\left(x-1\right)\left(x-4\right)+9x$$

x-1 ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-12x+9=x^{2}-5x+4+9x$$

4x ni olish uchun -5x va 9x ni birlashtirish.

$$4x^{2}-12x+9=x^{2}+4x+4$$

Ikkala tarafdan x^{2} ni ayirish.

$$4x^{2}-12x+9-x^{2}=4x+4$$

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

$$3x^{2}-12x+9=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$3x^{2}-12x+9-4x=4$$

-16x ni olish uchun -12x va -4x ni birlashtirish.

$$3x^{2}-16x+9=4$$

Ikkala tarafdan 9 ni ayirish.

$$3x^{2}-16x=4-9$$

-5 olish uchun 4 dan 9 ni ayirish.

$$3x^{2}-16x=-5$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-16x}{3}=-\frac{5}{3}$$

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

$$x^{2}-\frac{16}{3}x=-\frac{5}{3}$$

-\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{8}{3} olish uchun. Keyin, -\frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{16}{3}x+\left(-\frac{8}{3}\right)^{2}=-\frac{5}{3}+\left(-\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{8}{3} kvadratini chiqarish.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=-\frac{5}{3}+\frac{64}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{3} ni \frac{64}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{49}{9}$$

x^{2}-\frac{16}{3}x+\frac{64}{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.

$$\left(x-\frac{8}{3}\right)^{2}=\frac{49}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{8}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Qisqartirish.

$$x-\frac{8}{3}=\frac{7}{3} x-\frac{8}{3}=-\frac{7}{3}$$

\frac{8}{3} ni tenglamaning ikkala tarafiga qo'shish.

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