x^2/(4-x)(1-x)=13*10^9 eching

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 13 \times 10 ^ { 9 }$$

x uchun yechish

$$x = \frac{20000 \sqrt{950625000130} + 32500000000}{12999999999} \approx 4$$

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}\approx 1$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

9 daraja ko‘rsatkichini 10 ga hisoblang va 1000000000 ni qiymatni oling.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

13000000000 hosil qilish uchun 13 va 1000000000 ni ko'paytirish.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

13000000000 ga x-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

13000000000x-52000000000 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Ikkala tarafdan 13000000000x^{2} ni ayirish.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

-12999999999x^{2} ni olish uchun x^{2} va -13000000000x^{2} ni birlashtirish.

$$-12999999999x^{2}=-65000000000x+52000000000$$

65000000000x ni ikki tarafga qo’shing.

$$-12999999999x^{2}[phone]x=52000000000$$

Ikkala tarafdan 52000000000 ni ayirish.

$$-12999999999x^{2}[phone]x-52000000000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -12999999999 ni a, 65000000000 ni b va -52000000000 ni c bilan almashtiring.

$$x=\frac{-65000000000±\sqrt{65000000000^{2}-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

65000000000 kvadratini chiqarish.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

-4 ni -12999999999 marotabaga ko'paytirish.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000+51999999996\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

51999999996 ni -52000000000 marotabaga ko'paytirish.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-2703999999792000000000}}{2\left(-12999999999\right)}$$

4225000000000000000000 ni -2703999999792000000000 ga qo'shish.

$$x=\frac{-65000000000±\sqrt{1521000000208000000000}}{2\left(-12999999999\right)}$$

1521000000208000000000 ning kvadrat ildizini chiqarish.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{2\left(-12999999999\right)}$$

2 ni -12999999999 marotabaga ko'paytirish.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}$$

x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} tenglamasini yeching, bunda ± musbat. -65000000000 ni 40000\sqrt{950625000130} ga qo'shish.

$$x=\frac{40000\sqrt{950625000130}-65000000000}{-25999999998}$$

-65000000000+40000\sqrt{950625000130} ni -25999999998 ga bo'lish.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$

x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} tenglamasini yeching, bunda ± manfiy. -65000000000 dan 40000\sqrt{950625000130} ni ayirish.

$$x=\frac{-40000\sqrt{950625000130}-65000000000}{-25999999998}$$

-65000000000-40000\sqrt{950625000130} ni -25999999998 ga bo'lish.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Tenglama yechildi.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999} x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

9 daraja ko‘rsatkichini 10 ga hisoblang va 1000000000 ni qiymatni oling.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

13000000000 hosil qilish uchun 13 va 1000000000 ni ko'paytirish.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

13000000000 ga x-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

13000000000x-52000000000 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Ikkala tarafdan 13000000000x^{2} ni ayirish.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

-12999999999x^{2} ni olish uchun x^{2} va -13000000000x^{2} ni birlashtirish.

$$-12999999999x^{2}=-65000000000x+52000000000$$

65000000000x ni ikki tarafga qo’shing.

$$-12999999999x^{2}[phone]x=52000000000$$

Ikki tarafini -12999999999 ga bo‘ling.

$$\frac{-12999999999x^{2}[phone]x}{-12999999999}=\frac{52000000000}{-12999999999}$$

-12999999999 ga bo'lish -12999999999 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{65000000000}{-12999999999}x=\frac{52000000000}{-12999999999}$$

65000000000 ni -12999999999 ga bo'lish.

$$x^{2}-\frac{65000000000}{12999999999}x=\frac{52000000000}{-12999999999}$$

52000000000 ni -12999999999 ga bo'lish.

$$x^{2}-\frac{65000000000}{12999999999}x=-\frac{52000000000}{12999999999}$$

-\frac{65000000000}{12999999999} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{32500000000}{12999999999} olish uchun. Keyin, -\frac{32500000000}{12999999999} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{65000000000}{12999999999}x+\left(-\frac{32500000000}{12999999999}\right)^{2}=-\frac{52000000000}{12999999999}+\left(-\frac{32500000000}{12999999999}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{32500000000}{12999999999} kvadratini chiqarish.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=-\frac{52000000000}{12999999999}+\frac{1056250000000000000000}{168999999974000000001}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{52000000000}{12999999999} ni \frac{1056250000000000000000}{168999999974000000001} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=\frac{380250000052000000000}{168999999974000000001}$$

x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{32500000000}{12999999999}\right)^{2}=\frac{380250000052000000000}{168999999974000000001}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{32500000000}{12999999999}\right)^{2}}=\sqrt{\frac{380250000052000000000}{168999999974000000001}}$$

Qisqartirish.

$$x-\frac{32500000000}{12999999999}=\frac{20000\sqrt{950625000130}}{12999999999} x-\frac{32500000000}{12999999999}=-\frac{20000\sqrt{950625000130}}{12999999999}$$

\frac{32500000000}{12999999999} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999} x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$