(2x)^2/(1-x)(1-5-x)=12*10^-2 eching

$$\frac { ( 2 x ) ^ { 2 } } { ( 1 - x ) ( 1 - 5 - x ) } = 12 \times 10 ^ { - 2 }$$

x uchun yechish (complex solution)

$$x=\frac{9+5\sqrt{183}i}{194}\approx 0,046391753+0,348653331i$$

$$x=\frac{-5\sqrt{183}i+9}{194}\approx 0,046391753-0,348653331i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -4,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+4\right) ga ko'paytirish.

$$\left(2x\right)^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

\left(2x\right)^{2} ni kengaytirish.

$$2^{2}x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

-2 daraja ko‘rsatkichini 10 ga hisoblang va \frac{1}{100} ni qiymatni oling.

$$4x^{2}=12\times \frac{1}{100}\left(x-1\right)\left(x+4\right)$$

\frac{3}{25} hosil qilish uchun 12 va \frac{1}{100} ni ko'paytirish.

$$4x^{2}=\frac{3}{25}\left(x-1\right)\left(x+4\right)$$

\frac{3}{25} ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{3}{25}x-\frac{3}{25} ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}=\frac{3}{25}x^{2}+\frac{9}{25}x-\frac{12}{25}$$

Ikkala tarafdan \frac{3}{25}x^{2} ni ayirish.

$$4x^{2}-\frac{3}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

\frac{97}{25}x^{2} ni olish uchun 4x^{2} va -\frac{3}{25}x^{2} ni birlashtirish.

$$\frac{97}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Ikkala tarafdan \frac{9}{25}x ni ayirish.

$$\frac{97}{25}x^{2}-\frac{9}{25}x=-\frac{12}{25}$$

\frac{12}{25} ni ikki tarafga qo’shing.

$$\frac{97}{25}x^{2}-\frac{9}{25}x+\frac{12}{25}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{97}{25} ni a, -\frac{9}{25} ni b va \frac{12}{25} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\left(-\frac{9}{25}\right)^{2}-4\times \frac{97}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{25} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81}{625}-4\times \frac{97}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

-4 ni \frac{97}{25} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81}{625}-\frac{388}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{388}{25} ni \frac{12}{25} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81-4656}{625}}}{2\times \frac{97}{25}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{81}{625} ni -\frac{4656}{625} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{-\frac{183}{25}}}{2\times \frac{97}{25}}$$

-\frac{183}{25} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{9}{25}\right)±\frac{\sqrt{183}i}{5}}{2\times \frac{97}{25}}$$

-\frac{9}{25} ning teskarisi \frac{9}{25} ga teng.

$$x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{2\times \frac{97}{25}}$$

2 ni \frac{97}{25} marotabaga ko'paytirish.

$$x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}}$$

x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}} tenglamasini yeching, bunda ± musbat. \frac{9}{25} ni \frac{i\sqrt{183}}{5} ga qo'shish.

$$x=\frac{\frac{\sqrt{183}i}{5}+\frac{9}{25}}{\frac{194}{25}}$$

\frac{9}{25}+\frac{i\sqrt{183}}{5} ni \frac{194}{25} ga bo'lish \frac{9}{25}+\frac{i\sqrt{183}}{5} ga k'paytirish \frac{194}{25} ga qaytarish.

$$x=\frac{9+5\sqrt{183}i}{194}$$

x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}} tenglamasini yeching, bunda ± manfiy. \frac{9}{25} dan \frac{i\sqrt{183}}{5} ni ayirish.

$$x=\frac{-\frac{\sqrt{183}i}{5}+\frac{9}{25}}{\frac{194}{25}}$$

\frac{9}{25}-\frac{i\sqrt{183}}{5} ni \frac{194}{25} ga bo'lish \frac{9}{25}-\frac{i\sqrt{183}}{5} ga k'paytirish \frac{194}{25} ga qaytarish.

$$x=\frac{-5\sqrt{183}i+9}{194}$$

Tenglama yechildi.

$$x=\frac{9+5\sqrt{183}i}{194} x=\frac{-5\sqrt{183}i+9}{194}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -4,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+4\right) ga ko'paytirish.

$$\left(2x\right)^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

\left(2x\right)^{2} ni kengaytirish.

$$2^{2}x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

-2 daraja ko‘rsatkichini 10 ga hisoblang va \frac{1}{100} ni qiymatni oling.

$$4x^{2}=12\times \frac{1}{100}\left(x-1\right)\left(x+4\right)$$

\frac{3}{25} hosil qilish uchun 12 va \frac{1}{100} ni ko'paytirish.

$$4x^{2}=\frac{3}{25}\left(x-1\right)\left(x+4\right)$$

\frac{3}{25} ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{3}{25}x-\frac{3}{25} ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}=\frac{3}{25}x^{2}+\frac{9}{25}x-\frac{12}{25}$$

Ikkala tarafdan \frac{3}{25}x^{2} ni ayirish.

$$4x^{2}-\frac{3}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

\frac{97}{25}x^{2} ni olish uchun 4x^{2} va -\frac{3}{25}x^{2} ni birlashtirish.

$$\frac{97}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Ikkala tarafdan \frac{9}{25}x ni ayirish.

$$\frac{97}{25}x^{2}-\frac{9}{25}x=-\frac{12}{25}$$

Tenglamaning ikki tarafini \frac{97}{25} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{97}{25}x^{2}-\frac{9}{25}x}{\frac{97}{25}}=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

\frac{97}{25} ga bo'lish \frac{97}{25} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{9}{25}}{\frac{97}{25}}\right)x=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

-\frac{9}{25} ni \frac{97}{25} ga bo'lish -\frac{9}{25} ga k'paytirish \frac{97}{25} ga qaytarish.

$$x^{2}-\frac{9}{97}x=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

-\frac{12}{25} ni \frac{97}{25} ga bo'lish -\frac{12}{25} ga k'paytirish \frac{97}{25} ga qaytarish.

$$x^{2}-\frac{9}{97}x=-\frac{12}{97}$$

-\frac{9}{97} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{194} olish uchun. Keyin, -\frac{9}{194} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{97}x+\left(-\frac{9}{194}\right)^{2}=-\frac{12}{97}+\left(-\frac{9}{194}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{194} kvadratini chiqarish.

$$x^{2}-\frac{9}{97}x+\frac{81}{37636}=-\frac{12}{97}+\frac{81}{37636}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{12}{97} ni \frac{81}{37636} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{97}x+\frac{81}{37636}=-\frac{4575}{37636}$$

x^{2}-\frac{9}{97}x+\frac{81}{37636} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{194}\right)^{2}=-\frac{4575}{37636}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{194}\right)^{2}}=\sqrt{-\frac{4575}{37636}}$$

Qisqartirish.

$$x-\frac{9}{194}=\frac{5\sqrt{183}i}{194} x-\frac{9}{194}=-\frac{5\sqrt{183}i}{194}$$

\frac{9}{194} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9+5\sqrt{183}i}{194} x=\frac{-5\sqrt{183}i+9}{194}$$