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

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

x uchun yechish

$$x=-2$$

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

Kvadrat tenglamalaridan foydalanish qoidalari

3x ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

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

Ikkala tarafdan 2 ni ayirish.

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

2x ni ikki tarafga qo’shing.

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

5x ni olish uchun 3x va 2x ni birlashtirish.

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

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

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

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

-12 ni -2 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+24}}{2\times 3}$$

25 ni 24 ga qo'shish.

$$x=\frac{-5±\sqrt{49}}{2\times 3}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±7}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-5±7}{6}$$

x=\frac{-5±7}{6} tenglamasini yeching, bunda ± musbat. -5 ni 7 ga qo'shish.

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

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

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

x=\frac{-5±7}{6} tenglamasini yeching, bunda ± manfiy. -5 dan 7 ni ayirish.

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

-12 ni 6 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=\frac{1}{3} x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

3x ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

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

2x ni ikki tarafga qo’shing.

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

5x ni olish uchun 3x va 2x ni birlashtirish.

$$3x^{2}+5x=2$$

Ikki tarafini 3 ga bo‘ling.

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

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

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

\frac{5}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{6} olish uchun. Keyin, \frac{5}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{2}{3}+\left(\frac{5}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{6} kvadratini chiqarish.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{2}{3} ni \frac{25}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

x^{2}+\frac{5}{3}x+\frac{25}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{49}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Qisqartirish.

$$x+\frac{5}{6}=\frac{7}{6} x+\frac{5}{6}=-\frac{7}{6}$$

Tenglamaning ikkala tarafidan \frac{5}{6} ni ayirish.

$$x=\frac{1}{3} x=-2$$