8x+16+8/x+2*1/x-2=8x^2-25/x-2 eching

$$8 x + 16 + \frac { 8 } { x + 2 } \cdot \frac { 1 } { x - 2 } = \frac { 8 x ^ { 2 } - 25 } { x - 2 }$$

x uchun yechish

$$x=-\frac{6}{7}\approx -0,857142857$$

Guruhlash orqali faktorlash qadamlari

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

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x^{2}-16x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

x^{2}-4 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

\left(x-2\right)\times \frac{1}{x-2} ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

x+2 ga 8x^{2}-25 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

\frac{x-2}{x-2}\times 8 ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 8x^{3}-32x+16x^{2}-64 ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} va \frac{\left(x-2\right)\times 8}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8 ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16}{x-2}=8x^{3}-25x+16x^{2}-50$$

8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16 kabi iboralarga o‘xshab birlashtiring.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ikkala tarafdan 8x^{3} ni ayirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}-8x^{3}=-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -8x^{3} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

\frac{8x^{4}-64x^{2}+8x+112}{x-2} va \frac{-8x^{3}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}=-25x+16x^{2}-50$$

25x ni ikki tarafga qo’shing.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+25x=16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 25x ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

\frac{-64x^{2}+8x+112+16x^{3}}{x-2} va \frac{25x\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x^{2}-50x kabi iboralarga o‘xshab birlashtiring.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Ikkala tarafdan 16x^{2} ni ayirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -16x^{2} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

\frac{-39x^{2}-42x+112+16x^{3}}{x-2} va \frac{-16x^{2}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

50 ni ikki tarafga qo’shing.

$$\frac{-7x^{2}-42x+112}{x-2}+50=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 50 ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-7x^{2}-42x+112}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

\frac{-7x^{2}-42x+112}{x-2} va \frac{50\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-7x^{2}-42x+112+50\left(x-2\right)}{x-2}=0$$

-7x^{2}-42x+112+50\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-7x^{2}-42x+112+50x-100}{x-2}=0$$

-7x^{2}-42x+112+50x-100 kabi iboralarga o‘xshab birlashtiring.

$$\frac{-7x^{2}+8x+12}{x-2}=0$$

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$-7x^{2}+8x+12=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -7x^{2}+ax+bx+12 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=8 ab=-7\times 12=-84$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -84-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

Yechim – 8 yigʻindisini beruvchi juftlik.

$$a=14 b=-6$$

-7x^{2}+8x+12 ni \left(-7x^{2}+14x\right)+\left(-6x+12\right) sifatida qaytadan yozish.

$$\left(-7x^{2}+14x\right)+\left(-6x+12\right)$$

Birinchi guruhda 7x ni va ikkinchi guruhda 6 ni faktordan chiqaring.

$$7x\left(-x+2\right)+6\left(-x+2\right)$$

Distributiv funktsiyasidan foydalangan holda -x+2 umumiy terminini chiqaring.

$$\left(-x+2\right)\left(7x+6\right)$$

Tenglamani yechish uchun -x+2=0 va 7x+6=0 ni yeching.

$$x=2 x=-\frac{6}{7}$$

x qiymati 2 teng bo‘lmaydi.

$$x=-\frac{6}{7}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x^{2}-16x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

x^{2}-4 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

\left(x-2\right)\times \frac{1}{x-2} ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

x+2 ga 8x^{2}-25 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

\frac{x-2}{x-2}\times 8 ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 8x^{3}-32x+16x^{2}-64 ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} va \frac{\left(x-2\right)\times 8}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8 ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16}{x-2}=8x^{3}-25x+16x^{2}-50$$

8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16 kabi iboralarga o‘xshab birlashtiring.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ikkala tarafdan 8x^{3} ni ayirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}-8x^{3}=-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -8x^{3} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

\frac{8x^{4}-64x^{2}+8x+112}{x-2} va \frac{-8x^{3}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}=-25x+16x^{2}-50$$

25x ni ikki tarafga qo’shing.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+25x=16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 25x ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

\frac{-64x^{2}+8x+112+16x^{3}}{x-2} va \frac{25x\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x^{2}-50x kabi iboralarga o‘xshab birlashtiring.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Ikkala tarafdan 16x^{2} ni ayirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -16x^{2} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

\frac{-39x^{2}-42x+112+16x^{3}}{x-2} va \frac{-16x^{2}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

50 ni ikki tarafga qo’shing.

$$\frac{-7x^{2}-42x+112}{x-2}+50=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 50 ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-7x^{2}-42x+112}{x-2}+\frac{50\left(x-2\right)}{x-2}=0$$

\frac{-7x^{2}-42x+112}{x-2} va \frac{50\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-7x^{2}-42x+112+50\left(x-2\right)}{x-2}=0$$

-7x^{2}-42x+112+50\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-7x^{2}-42x+112+50x-100}{x-2}=0$$

-7x^{2}-42x+112+50x-100 kabi iboralarga o‘xshab birlashtiring.

$$\frac{-7x^{2}+8x+12}{x-2}=0$$

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$-7x^{2}+8x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -7 ni a, 8 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-7\right)\times 12}}{2\left(-7\right)}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\left(-7\right)\times 12}}{2\left(-7\right)}$$

-4 ni -7 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64+28\times 12}}{2\left(-7\right)}$$

28 ni 12 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64+336}}{2\left(-7\right)}$$

64 ni 336 ga qo'shish.

$$x=\frac{-8±\sqrt{400}}{2\left(-7\right)}$$

400 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±20}{2\left(-7\right)}$$

2 ni -7 marotabaga ko'paytirish.

$$x=\frac{-8±20}{-14}$$

x=\frac{-8±20}{-14} tenglamasini yeching, bunda ± musbat. -8 ni 20 ga qo'shish.

$$x=\frac{12}{-14}$$

\frac{12}{-14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{6}{7}$$

x=\frac{-8±20}{-14} tenglamasini yeching, bunda ± manfiy. -8 dan 20 ni ayirish.

$$x=-\frac{28}{-14}$$

-28 ni -14 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=-\frac{6}{7} x=2$$

x qiymati 2 teng bo‘lmaydi.

$$x=-\frac{6}{7}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$8x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(8x^{2}-16x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

8x^{2}-16x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{3}-32x+\left(x-2\right)\left(x+2\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

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

$$8x^{3}-32x+\left(x^{2}-4\right)\times 16+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

x^{2}-4 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\left(x-2\right)\times 8\times \frac{1}{x-2}=\left(x+2\right)\left(8x^{2}-25\right)$$

\left(x-2\right)\times \frac{1}{x-2} ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=\left(x+2\right)\left(8x^{2}-25\right)$$

x+2 ga 8x^{2}-25 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{3}-32x+16x^{2}-64+\frac{x-2}{x-2}\times 8=8x^{3}-25x+16x^{2}-50$$

\frac{x-2}{x-2}\times 8 ni yagona kasrga aylantiring.

$$8x^{3}-32x+16x^{2}-64+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 8x^{3}-32x+16x^{2}-64 ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2}+\frac{\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)}{x-2} va \frac{\left(x-2\right)\times 8}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8}{x-2}=8x^{3}-25x+16x^{2}-50$$

\left(8x^{3}-32x+16x^{2}-64\right)\left(x-2\right)+\left(x-2\right)\times 8 ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16}{x-2}=8x^{3}-25x+16x^{2}-50$$

8x^{4}-16x^{3}-32x^{2}+64x+16x^{3}-32x^{2}-64x+128+8x-16 kabi iboralarga o‘xshab birlashtiring.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}=8x^{3}-25x+16x^{2}-50$$

Ikkala tarafdan 8x^{3} ni ayirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}-8x^{3}=-25x+16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -8x^{3} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{8x^{4}-64x^{2}+8x+112}{x-2}+\frac{-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

\frac{8x^{4}-64x^{2}+8x+112}{x-2} va \frac{-8x^{3}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right)}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{3}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3}}{x-2}=-25x+16x^{2}-50$$

8x^{4}-64x^{2}+8x+112-8x^{4}+16x^{3} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}=-25x+16x^{2}-50$$

25x ni ikki tarafga qo’shing.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+25x=16x^{2}-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 25x ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-64x^{2}+8x+112+16x^{3}}{x-2}+\frac{25x\left(x-2\right)}{x-2}=16x^{2}-50$$

\frac{-64x^{2}+8x+112+16x^{3}}{x-2} va \frac{25x\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right)}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-64x^{2}+8x+112+16x^{3}+25x^{2}-50x}{x-2}=16x^{2}-50$$

-64x^{2}+8x+112+16x^{3}+25x^{2}-50x kabi iboralarga o‘xshab birlashtiring.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}=16x^{2}-50$$

Ikkala tarafdan 16x^{2} ni ayirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}-16x^{2}=-50$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -16x^{2} ni \frac{x-2}{x-2} marotabaga ko'paytirish.

$$\frac{-39x^{2}-42x+112+16x^{3}}{x-2}+\frac{-16x^{2}\left(x-2\right)}{x-2}=-50$$

\frac{-39x^{2}-42x+112+16x^{3}}{x-2} va \frac{-16x^{2}\left(x-2\right)}{x-2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right)}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{2}\left(x-2\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2}}{x-2}=-50$$

-39x^{2}-42x+112+16x^{3}-16x^{3}+32x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{-7x^{2}-42x+112}{x-2}=-50$$

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$-7x^{2}-42x+112=-50\left(x-2\right)$$

-50 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-7x^{2}-42x+112=-50x+100$$

50x ni ikki tarafga qo’shing.

$$-7x^{2}-42x+112+50x=100$$

8x ni olish uchun -42x va 50x ni birlashtirish.

$$-7x^{2}+8x+112=100$$

Ikkala tarafdan 112 ni ayirish.

$$-7x^{2}+8x=100-112$$

-12 olish uchun 100 dan 112 ni ayirish.

$$-7x^{2}+8x=-12$$

Ikki tarafini -7 ga bo‘ling.

$$\frac{-7x^{2}+8x}{-7}=-\frac{12}{-7}$$

-7 ga bo'lish -7 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{8}{-7}x=-\frac{12}{-7}$$

8 ni -7 ga bo'lish.

$$x^{2}-\frac{8}{7}x=-\frac{12}{-7}$$

-12 ni -7 ga bo'lish.

$$x^{2}-\frac{8}{7}x=\frac{12}{7}$$

-\frac{8}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{7} olish uchun. Keyin, -\frac{4}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{7}x+\left(-\frac{4}{7}\right)^{2}=\frac{12}{7}+\left(-\frac{4}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{7} kvadratini chiqarish.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{12}{7}+\frac{16}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{12}{7} ni \frac{16}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{7}x+\frac{16}{49}=\frac{100}{49}$$

x^{2}-\frac{8}{7}x+\frac{16}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{7}\right)^{2}=\frac{100}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{7}\right)^{2}}=\sqrt{\frac{100}{49}}$$

Qisqartirish.

$$x-\frac{4}{7}=\frac{10}{7} x-\frac{4}{7}=-\frac{10}{7}$$

\frac{4}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=-\frac{6}{7}$$

x qiymati 2 teng bo‘lmaydi.

$$x=-\frac{6}{7}$$