x(2x+1)-(x-1)^2/2=3 eching

$$x ( 2 x + 1 ) - \frac { ( x - 1 ) ^ { 2 } } { 2 } = 3$$

x uchun yechish

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2,333333333$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

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

2x ga 2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+2x-\left(x-1\right)^{2}=6$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+2x-\left(x^{2}-2x+1\right)=6$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

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

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

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

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

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

Ikkala tarafdan 6 ni ayirish.

$$3x^{2}+4x-1-6=0$$

-7 olish uchun -1 dan 6 ni ayirish.

$$3x^{2}+4x-7=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-7 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=4 ab=3\left(-7\right)=-21$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -21-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,21 -3,7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+21=20 -3+7=4$$

Yechim – 4 yigʻindisini beruvchi juftlik.

$$a=-3 b=7$$

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

$$\left(3x^{2}-3x\right)+\left(7x-7\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 7 ni faktordan chiqaring.

$$3x\left(x-1\right)+7\left(x-1\right)$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(3x+7\right)$$

Tenglamani yechish uchun x-1=0 va 3x+7=0 ni yeching.

$$x=1 x=-\frac{7}{3}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

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

2x ga 2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+2x-\left(x-1\right)^{2}=6$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+2x-\left(x^{2}-2x+1\right)=6$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

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

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

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

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

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

Ikkala tarafdan 6 ni ayirish.

$$3x^{2}+4x-1-6=0$$

-7 olish uchun -1 dan 6 ni ayirish.

$$3x^{2}+4x-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 4 ni b va -7 ni c bilan almashtiring.

$$x=\frac{-4±\sqrt{4^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\times 3\left(-7\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-12\left(-7\right)}}{2\times 3}$$

-12 ni -7 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+84}}{2\times 3}$$

16 ni 84 ga qo'shish.

$$x=\frac{-4±\sqrt{100}}{2\times 3}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±10}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-4±10}{6}$$

x=\frac{-4±10}{6} tenglamasini yeching, bunda ± musbat. -4 ni 10 ga qo'shish.

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

6 ni 6 ga bo'lish.

$$x=1$$

x=\frac{-4±10}{6} tenglamasini yeching, bunda ± manfiy. -4 dan 10 ni ayirish.

$$x=-\frac{14}{6}$$

\frac{-14}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=1 x=-\frac{7}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

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

2x ga 2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+2x-\left(x-1\right)^{2}=6$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+2x-\left(x^{2}-2x+1\right)=6$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

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

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

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

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

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

1 ni ikki tarafga qo’shing.

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

7 olish uchun 6 va 1'ni qo'shing.

$$3x^{2}+4x=7$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+4x}{3}=\frac{7}{3}$$

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

$$x^{2}+\frac{4}{3}x=\frac{7}{3}$$

\frac{4}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{2}{3} olish uchun. Keyin, \frac{2}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=\frac{7}{3}+\left(\frac{2}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{2}{3} kvadratini chiqarish.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{7}{3}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{3} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{25}{9}$$

x^{2}+\frac{4}{3}x+\frac{4}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+\frac{2}{3}=\frac{5}{3} x+\frac{2}{3}=-\frac{5}{3}$$

Tenglamaning ikkala tarafidan \frac{2}{3} ni ayirish.

$$x=1 x=-\frac{7}{3}$$