1/6(4x+5)-2/3(2x+7)=3 eching

$$\frac{ 1 }{ 6 } (4x+5) \frac{ -2 }{ 3 } (2x+7)=3$$

x uchun yechish (complex solution)

$$x=\frac{-19+3\sqrt{15}i}{8}\approx -2,375+1,452368755i$$

$$x=\frac{-3\sqrt{15}i-19}{8}\approx -2,375-1,452368755i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{-2}{3} kasri manfiy belgini olib tashlash bilan -\frac{2}{3} sifatida qayta yozilishi mumkin.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

-\frac{1}{9} hosil qilish uchun \frac{1}{6} va -\frac{2}{3} ni ko'paytirish.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

-\frac{1}{9} ga 4x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

-\frac{4}{9}x-\frac{5}{9} ga 2x+7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

Ikkala tarafdan 3 ni ayirish.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}-3=0$$

-\frac{62}{9} olish uchun -\frac{35}{9} dan 3 ni ayirish.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{62}{9}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{8}{9} ni a, -\frac{38}{9} ni b va -\frac{62}{9} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\left(-\frac{38}{9}\right)^{2}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{38}{9} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

-4 ni -\frac{8}{9} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}+\frac{32}{9}\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{32}{9} ni -\frac{62}{9} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444-1984}{81}}}{2\left(-\frac{8}{9}\right)}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1444}{81} ni -\frac{1984}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{-\frac{20}{3}}}{2\left(-\frac{8}{9}\right)}$$

-\frac{20}{3} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{38}{9}\right)±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

-\frac{38}{9} ning teskarisi \frac{38}{9} ga teng.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

2 ni -\frac{8}{9} marotabaga ko'paytirish.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}}$$

x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} tenglamasini yeching, bunda ± musbat. \frac{38}{9} ni \frac{2i\sqrt{15}}{3} ga qo'shish.

$$x=\frac{\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

\frac{38}{9}+\frac{2i\sqrt{15}}{3} ni -\frac{16}{9} ga bo'lish \frac{38}{9}+\frac{2i\sqrt{15}}{3} ga k'paytirish -\frac{16}{9} ga qaytarish.

$$x=\frac{-3\sqrt{15}i-19}{8}$$

x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} tenglamasini yeching, bunda ± manfiy. \frac{38}{9} dan \frac{2i\sqrt{15}}{3} ni ayirish.

$$x=\frac{-\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

\frac{38}{9}-\frac{2i\sqrt{15}}{3} ni -\frac{16}{9} ga bo'lish \frac{38}{9}-\frac{2i\sqrt{15}}{3} ga k'paytirish -\frac{16}{9} ga qaytarish.

$$x=\frac{-19+3\sqrt{15}i}{8}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{15}i-19}{8} x=\frac{-19+3\sqrt{15}i}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{-2}{3} kasri manfiy belgini olib tashlash bilan -\frac{2}{3} sifatida qayta yozilishi mumkin.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

-\frac{1}{9} hosil qilish uchun \frac{1}{6} va -\frac{2}{3} ni ko'paytirish.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

-\frac{1}{9} ga 4x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

-\frac{4}{9}x-\frac{5}{9} ga 2x+7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

\frac{35}{9} ni ikki tarafga qo’shing.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=3+\frac{35}{9}$$

\frac{62}{9} olish uchun 3 va \frac{35}{9}'ni qo'shing.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=\frac{62}{9}$$

Tenglamaning ikki tarafini -\frac{8}{9} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{-\frac{8}{9}x^{2}-\frac{38}{9}x}{-\frac{8}{9}}=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

-\frac{8}{9} ga bo'lish -\frac{8}{9} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{38}{9}}{-\frac{8}{9}}\right)x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

-\frac{38}{9} ni -\frac{8}{9} ga bo'lish -\frac{38}{9} ga k'paytirish -\frac{8}{9} ga qaytarish.

$$x^{2}+\frac{19}{4}x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

\frac{62}{9} ni -\frac{8}{9} ga bo'lish \frac{62}{9} ga k'paytirish -\frac{8}{9} ga qaytarish.

$$x^{2}+\frac{19}{4}x=-\frac{31}{4}$$

\frac{19}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{19}{8} olish uchun. Keyin, \frac{19}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{19}{4}x+\left(\frac{19}{8}\right)^{2}=-\frac{31}{4}+\left(\frac{19}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{19}{8} kvadratini chiqarish.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{31}{4}+\frac{361}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{31}{4} ni \frac{361}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{135}{64}$$

x^{2}+\frac{19}{4}x+\frac{361}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{19}{8}\right)^{2}=-\frac{135}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{19}{8}\right)^{2}}=\sqrt{-\frac{135}{64}}$$

Qisqartirish.

$$x+\frac{19}{8}=\frac{3\sqrt{15}i}{8} x+\frac{19}{8}=-\frac{3\sqrt{15}i}{8}$$

Tenglamaning ikkala tarafidan \frac{19}{8} ni ayirish.

$$x=\frac{-19+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i-19}{8}$$