(x-1)(2x+3)+(x-1)(5x-2)=0 eching

$$(x-1)(2x+3)+(x-1)(5x-2)=0$$

x uchun yechish

$$x=-\frac{1}{7}\approx -0,142857143$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

x-1 ga 2x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

x-1 ga 5x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

7x^{2} ni olish uchun 2x^{2} va 5x^{2} ni birlashtirish.

$$7x^{2}+x-3-7x+2=0$$

-6x ni olish uchun x va -7x ni birlashtirish.

$$7x^{2}-6x-3+2=0$$

-1 olish uchun -3 va 2'ni qo'shing.

$$7x^{2}-6x-1=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 7x^{2}+ax+bx-1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-6 ab=7\left(-1\right)=-7$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=-7 b=1$$

7x^{2}-6x-1 ni \left(7x^{2}-7x\right)+\left(x-1\right) sifatida qaytadan yozish.

$$\left(7x^{2}-7x\right)+\left(x-1\right)$$

7x^{2}-7x ichida 7x ni ajrating.

$$7x\left(x-1\right)+x-1$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(7x+1\right)$$

Tenglamani yechish uchun x-1=0 va 7x+1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

x-1 ga 2x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

x-1 ga 5x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

7x^{2} ni olish uchun 2x^{2} va 5x^{2} ni birlashtirish.

$$7x^{2}+x-3-7x+2=0$$

-6x ni olish uchun x va -7x ni birlashtirish.

$$7x^{2}-6x-3+2=0$$

-1 olish uchun -3 va 2'ni qo'shing.

$$7x^{2}-6x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -6 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 7\left(-1\right)}}{2\times 7}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 7\left(-1\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-28\left(-1\right)}}{2\times 7}$$

-28 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+28}}{2\times 7}$$

36 ni 28 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{64}}{2\times 7}$$

64 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±8}{2\times 7}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±8}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{6±8}{14}$$

x=\frac{6±8}{14} tenglamasini yeching, bunda ± musbat. 6 ni 8 ga qo'shish.

$$x=\frac{14}{14}$$

14 ni 14 ga bo'lish.

$$x=1$$

x=\frac{6±8}{14} tenglamasini yeching, bunda ± manfiy. 6 dan 8 ni ayirish.

$$x=-\frac{2}{14}$$

\frac{-2}{14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

x-1 ga 2x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

x-1 ga 5x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

7x^{2} ni olish uchun 2x^{2} va 5x^{2} ni birlashtirish.

$$7x^{2}+x-3-7x+2=0$$

-6x ni olish uchun x va -7x ni birlashtirish.

$$7x^{2}-6x-3+2=0$$

-1 olish uchun -3 va 2'ni qo'shing.

$$7x^{2}-6x-1=0$$

1 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$7x^{2}-6x=1$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-6x}{7}=\frac{1}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{6}{7}x=\frac{1}{7}$$

-\frac{6}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{7} olish uchun. Keyin, -\frac{3}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{6}{7}x+\left(-\frac{3}{7}\right)^{2}=\frac{1}{7}+\left(-\frac{3}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{7} kvadratini chiqarish.

$$x^{2}-\frac{6}{7}x+\frac{9}{49}=\frac{1}{7}+\frac{9}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{7} ni \frac{9}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{6}{7}x+\frac{9}{49}=\frac{16}{49}$$

x^{2}-\frac{6}{7}x+\frac{9}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{7}\right)^{2}=\frac{16}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{7}\right)^{2}}=\sqrt{\frac{16}{49}}$$

Qisqartirish.

$$x-\frac{3}{7}=\frac{4}{7} x-\frac{3}{7}=-\frac{4}{7}$$

\frac{3}{7} ni tenglamaning ikkala tarafiga qo'shish.

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