3x(x-1)+4x=3/4(x+1)-6x eching

$$3 x ( x - 1 ) + 4 x = \frac { 3 } { 4 } ( x + 1 ) - 6 x$$

x uchun yechish

$$x=\frac{\sqrt{769}-25}{24}\approx 0,113785385$$

$$x=\frac{-\sqrt{769}-25}{24}\approx -2,197118719$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$3x^{2}-3x+4x=\frac{3}{4}\left(x+1\right)-6x$$

x ni olish uchun -3x va 4x ni birlashtirish.

$$3x^{2}+x=\frac{3}{4}\left(x+1\right)-6x$$

\frac{3}{4} ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

-\frac{21}{4}x ni olish uchun \frac{3}{4}x va -6x ni birlashtirish.

$$3x^{2}+x=-\frac{21}{4}x+\frac{3}{4}$$

\frac{21}{4}x ni ikki tarafga qo’shing.

$$3x^{2}+x+\frac{21}{4}x=\frac{3}{4}$$

\frac{25}{4}x ni olish uchun x va \frac{21}{4}x ni birlashtirish.

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

Ikkala tarafdan \frac{3}{4} ni ayirish.

$$3x^{2}+\frac{25}{4}x-\frac{3}{4}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, \frac{25}{4} ni b va -\frac{3}{4} ni c bilan almashtiring.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{25}{4} kvadratini chiqarish.

$$x=\frac{-\frac{25}{4}±\sqrt{\frac{625}{16}-4\times 3\left(-\frac{3}{4}\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\frac{25}{4}±\sqrt{\frac{625}{16}-12\left(-\frac{3}{4}\right)}}{2\times 3}$$

-12 ni -\frac{3}{4} marotabaga ko'paytirish.

$$x=\frac{-\frac{25}{4}±\sqrt{\frac{625}{16}+9}}{2\times 3}$$

\frac{625}{16} ni 9 ga qo'shish.

$$x=\frac{-\frac{25}{4}±\sqrt{\frac{769}{16}}}{2\times 3}$$

\frac{769}{16} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{25}{4}±\frac{\sqrt{769}}{4}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\frac{25}{4}±\frac{\sqrt{769}}{4}}{6}$$

x=\frac{-\frac{25}{4}±\frac{\sqrt{769}}{4}}{6} tenglamasini yeching, bunda ± musbat. -\frac{25}{4} ni \frac{\sqrt{769}}{4} ga qo'shish.

$$x=\frac{\sqrt{769}-25}{4\times 6}$$

\frac{-25+\sqrt{769}}{4} ni 6 ga bo'lish.

$$x=\frac{\sqrt{769}-25}{24}$$

x=\frac{-\frac{25}{4}±\frac{\sqrt{769}}{4}}{6} tenglamasini yeching, bunda ± manfiy. -\frac{25}{4} dan \frac{\sqrt{769}}{4} ni ayirish.

$$x=\frac{-\sqrt{769}-25}{4\times 6}$$

\frac{-25-\sqrt{769}}{4} ni 6 ga bo'lish.

$$x=\frac{-\sqrt{769}-25}{24}$$

Tenglama yechildi.

$$x=\frac{\sqrt{769}-25}{24} x=\frac{-\sqrt{769}-25}{24}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$3x^{2}-3x+4x=\frac{3}{4}\left(x+1\right)-6x$$

x ni olish uchun -3x va 4x ni birlashtirish.

$$3x^{2}+x=\frac{3}{4}\left(x+1\right)-6x$$

\frac{3}{4} ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

-\frac{21}{4}x ni olish uchun \frac{3}{4}x va -6x ni birlashtirish.

$$3x^{2}+x=-\frac{21}{4}x+\frac{3}{4}$$

\frac{21}{4}x ni ikki tarafga qo’shing.

$$3x^{2}+x+\frac{21}{4}x=\frac{3}{4}$$

\frac{25}{4}x ni olish uchun x va \frac{21}{4}x ni birlashtirish.

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

Ikki tarafini 3 ga bo‘ling.

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

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

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

\frac{25}{4} ni 3 ga bo'lish.

$$x^{2}+\frac{25}{12}x=\frac{\frac{3}{4}}{3}$$

\frac{3}{4} ni 3 ga bo'lish.

$$x^{2}+\frac{25}{12}x=\frac{1}{4}$$

\frac{25}{12} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{25}{24} olish uchun. Keyin, \frac{25}{24} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{25}{12}x+\left(\frac{25}{24}\right)^{2}=\frac{1}{4}+\left(\frac{25}{24}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{25}{24} kvadratini chiqarish.

$$x^{2}+\frac{25}{12}x+\frac{625}{576}=\frac{1}{4}+\frac{625}{576}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{4} ni \frac{625}{576} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{25}{12}x+\frac{625}{576}=\frac{769}{576}$$

x^{2}+\frac{25}{12}x+\frac{625}{576} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{25}{24}\right)^{2}=\frac{769}{576}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{25}{24}\right)^{2}}=\sqrt{\frac{769}{576}}$$

Qisqartirish.

$$x+\frac{25}{24}=\frac{\sqrt{769}}{24} x+\frac{25}{24}=-\frac{\sqrt{769}}{24}$$

Tenglamaning ikkala tarafidan \frac{25}{24} ni ayirish.

$$x=\frac{\sqrt{769}-25}{24} x=\frac{-\sqrt{769}-25}{24}$$