(2x-3)(4x-2)-x(2x-3)=0 eching

$$( 2 x - 3 ) ( 4 x - 2 ) - x ( 2 x - 3 ) = 0$$

x uchun yechish

$$x=\frac{2}{3}\approx 0,666666667$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1,5$$

Guruhlash orqali faktorlash qadamlari

2x-3 ga 4x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

2x^{2}-3x teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

6x^{2} ni olish uchun 8x^{2} va -2x^{2} ni birlashtirish.

$$6x^{2}-16x+6+3x=0$$

-13x ni olish uchun -16x va 3x ni birlashtirish.

$$6x^{2}-13x+6=0$$

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

$$a+b=-13 ab=6\times 6=36$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 36-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Yechim – -13 yigʻindisini beruvchi juftlik.

$$a=-9 b=-4$$

6x^{2}-13x+6 ni \left(6x^{2}-9x\right)+\left(-4x+6\right) sifatida qaytadan yozish.

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

Birinchi guruhda 3x ni va ikkinchi guruhda -2 ni faktordan chiqaring.

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

Distributiv funktsiyasidan foydalangan holda 2x-3 umumiy terminini chiqaring.

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

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

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

Kvadrat tenglamalaridan foydalanish qoidalari

2x-3 ga 4x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

2x^{2}-3x teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

6x^{2} ni olish uchun 8x^{2} va -2x^{2} ni birlashtirish.

$$6x^{2}-16x+6+3x=0$$

-13x ni olish uchun -16x va 3x ni birlashtirish.

$$6x^{2}-13x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -13 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 6}}{2\times 6}$$

-13 kvadratini chiqarish.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\times 6}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\times 6}}{2\times 6}$$

-24 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-13\right)±\sqrt{169-144}}{2\times 6}$$

169 ni -144 ga qo'shish.

$$x=\frac{-\left(-13\right)±\sqrt{25}}{2\times 6}$$

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-13\right)±5}{2\times 6}$$

-13 ning teskarisi 13 ga teng.

$$x=\frac{13±5}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{13±5}{12}$$

x=\frac{13±5}{12} tenglamasini yeching, bunda ± musbat. 13 ni 5 ga qo'shish.

$$x=\frac{18}{12}$$

\frac{18}{12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{13±5}{12} tenglamasini yeching, bunda ± manfiy. 13 dan 5 ni ayirish.

$$x=\frac{8}{12}$$

\frac{8}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

2x-3 ga 4x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-16x+6-x\left(2x-3\right)=0$$

x ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-16x+6-\left(2x^{2}-3x\right)=0$$

2x^{2}-3x teskarisini topish uchun har birining teskarisini toping.

$$8x^{2}-16x+6-2x^{2}+3x=0$$

6x^{2} ni olish uchun 8x^{2} va -2x^{2} ni birlashtirish.

$$6x^{2}-16x+6+3x=0$$

-13x ni olish uchun -16x va 3x ni birlashtirish.

$$6x^{2}-13x+6=0$$

Ikkala tarafdan 6 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$6x^{2}-13x=-6$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-13x}{6}=-\frac{6}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{13}{6}x=-\frac{6}{6}$$

-6 ni 6 ga bo'lish.

$$x^{2}-\frac{13}{6}x=-1$$

-\frac{13}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{12} olish uchun. Keyin, -\frac{13}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-1+\left(-\frac{13}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{12} kvadratini chiqarish.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-1+\frac{169}{144}$$

-1 ni \frac{169}{144} ga qo'shish.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{25}{144}$$

x^{2}-\frac{13}{6}x+\frac{169}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{25}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Qisqartirish.

$$x-\frac{13}{12}=\frac{5}{12} x-\frac{13}{12}=-\frac{5}{12}$$

\frac{13}{12} ni tenglamaning ikkala tarafiga qo'shish.

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