quadtext{17}x-3/x+3+x+3/x-3=21/2 eching

$$\quad \text { 17 } \frac { x - 3 } { x + 3 } + \frac { x + 3 } { x - 3 } = 2 \frac { 1 } { 2 }$$

x uchun yechish (complex solution)

$$x=\frac{-3\sqrt{247}i+96}{31}\approx 3,096774194-1,520925837i$$

$$x=\frac{96+3\sqrt{247}i}{31}\approx 3,096774194+1,520925837i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x-3\right)\left(x+3\right) ga, x+3,x-3,2 ning eng kichik karralisiga ko‘paytiring.

$$17\left(2x-6\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

17 ga 2x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(34x-102\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

34x-102 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$34x^{2}-204x+306+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

2x+6 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$34x^{2}-204x+306+2x^{2}+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

36x^{2} ni olish uchun 34x^{2} va 2x^{2} ni birlashtirish.

$$36x^{2}-204x+306+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

-192x ni olish uchun -204x va 12x ni birlashtirish.

$$36x^{2}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

324 olish uchun 306 va 18'ni qo'shing.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

4 hosil qilish uchun 2 va 2 ni ko'paytirish.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(4+1\right)$$

5 olish uchun 4 va 1'ni qo'shing.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\times 5$$

x^{2}-9 ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x^{2}-192x+324=5x^{2}-45$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$36x^{2}-192x+324-5x^{2}=-45$$

31x^{2} ni olish uchun 36x^{2} va -5x^{2} ni birlashtirish.

$$31x^{2}-192x+324=-45$$

45 ni ikki tarafga qo’shing.

$$31x^{2}-192x+324+45=0$$

369 olish uchun 324 va 45'ni qo'shing.

$$31x^{2}-192x+369=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 31 ni a, -192 ni b va 369 ni c bilan almashtiring.

$$x=\frac{-\left(-192\right)±\sqrt{\left(-192\right)^{2}-4\times 31\times 369}}{2\times 31}$$

-192 kvadratini chiqarish.

$$x=\frac{-\left(-192\right)±\sqrt{36864-4\times 31\times 369}}{2\times 31}$$

-4 ni 31 marotabaga ko'paytirish.

$$x=\frac{-\left(-192\right)±\sqrt{36864-124\times 369}}{2\times 31}$$

-124 ni 369 marotabaga ko'paytirish.

$$x=\frac{-\left(-192\right)±\sqrt{36864-45756}}{2\times 31}$$

36864 ni -45756 ga qo'shish.

$$x=\frac{-\left(-192\right)±\sqrt{-8892}}{2\times 31}$$

-8892 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-192\right)±6\sqrt{247}i}{2\times 31}$$

-192 ning teskarisi 192 ga teng.

$$x=\frac{192±6\sqrt{247}i}{2\times 31}$$

2 ni 31 marotabaga ko'paytirish.

$$x=\frac{192±6\sqrt{247}i}{62}$$

x=\frac{192±6\sqrt{247}i}{62} tenglamasini yeching, bunda ± musbat. 192 ni 6i\sqrt{247} ga qo'shish.

$$x=\frac{192+6\sqrt{247}i}{62}$$

192+6i\sqrt{247} ni 62 ga bo'lish.

$$x=\frac{96+3\sqrt{247}i}{31}$$

x=\frac{192±6\sqrt{247}i}{62} tenglamasini yeching, bunda ± manfiy. 192 dan 6i\sqrt{247} ni ayirish.

$$x=\frac{-6\sqrt{247}i+192}{62}$$

192-6i\sqrt{247} ni 62 ga bo'lish.

$$x=\frac{-3\sqrt{247}i+96}{31}$$

Tenglama yechildi.

$$x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x-3\right)\left(x+3\right) ga, x+3,x-3,2 ning eng kichik karralisiga ko‘paytiring.

$$17\left(2x-6\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

17 ga 2x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(34x-102\right)\left(x-3\right)+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

34x-102 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$34x^{2}-204x+306+\left(2x+6\right)\left(x+3\right)=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

2x+6 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$34x^{2}-204x+306+2x^{2}+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

36x^{2} ni olish uchun 34x^{2} va 2x^{2} ni birlashtirish.

$$36x^{2}-204x+306+12x+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

-192x ni olish uchun -204x va 12x ni birlashtirish.

$$36x^{2}-192x+306+18=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

324 olish uchun 306 va 18'ni qo'shing.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(2\times 2+1\right)$$

4 hosil qilish uchun 2 va 2 ni ko'paytirish.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\left(4+1\right)$$

5 olish uchun 4 va 1'ni qo'shing.

$$36x^{2}-192x+324=\left(x^{2}-9\right)\times 5$$

x^{2}-9 ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x^{2}-192x+324=5x^{2}-45$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$36x^{2}-192x+324-5x^{2}=-45$$

31x^{2} ni olish uchun 36x^{2} va -5x^{2} ni birlashtirish.

$$31x^{2}-192x+324=-45$$

Ikkala tarafdan 324 ni ayirish.

$$31x^{2}-192x=-45-324$$

-369 olish uchun -45 dan 324 ni ayirish.

$$31x^{2}-192x=-369$$

Ikki tarafini 31 ga bo‘ling.

$$\frac{31x^{2}-192x}{31}=-\frac{369}{31}$$

31 ga bo'lish 31 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{192}{31}x=-\frac{369}{31}$$

-\frac{192}{31} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{96}{31} olish uchun. Keyin, -\frac{96}{31} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{192}{31}x+\left(-\frac{96}{31}\right)^{2}=-\frac{369}{31}+\left(-\frac{96}{31}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{96}{31} kvadratini chiqarish.

$$x^{2}-\frac{192}{31}x+\frac{9216}{961}=-\frac{369}{31}+\frac{9216}{961}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{369}{31} ni \frac{9216}{961} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{192}{31}x+\frac{9216}{961}=-\frac{2223}{961}$$

x^{2}-\frac{192}{31}x+\frac{9216}{961} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{96}{31}\right)^{2}=-\frac{2223}{961}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{96}{31}\right)^{2}}=\sqrt{-\frac{2223}{961}}$$

Qisqartirish.

$$x-\frac{96}{31}=\frac{3\sqrt{247}i}{31} x-\frac{96}{31}=-\frac{3\sqrt{247}i}{31}$$

\frac{96}{31} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{96+3\sqrt{247}i}{31} x=\frac{-3\sqrt{247}i+96}{31}$$