x-2/x+3-30/x^2-9=3/8 eching

$$\frac { x - 2 } { x + 3 } - \frac { 30 } { x ^ { 2 } - 9 } = \frac { 3 } { 8 }$$

x uchun yechish

$$x=11$$

Guruhlash orqali faktorlash qadamlari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 8\left(x-3\right)\left(x+3\right) ga, x+3,x^{2}-9,8 ning eng kichik karralisiga ko‘paytiring.

$$\left(8x-24\right)\left(x-2\right)-8\times 30=3\left(x-3\right)\left(x+3\right)$$

8x-24 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x+48-8\times 30=3\left(x-3\right)\left(x+3\right)$$

-240 hosil qilish uchun -8 va 30 ni ko'paytirish.

$$8x^{2}-40x+48-240=3\left(x-3\right)\left(x+3\right)$$

-192 olish uchun 48 dan 240 ni ayirish.

$$8x^{2}-40x-192=3\left(x-3\right)\left(x+3\right)$$

3 ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-40x-192=\left(3x-9\right)\left(x+3\right)$$

3x-9 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x-192=3x^{2}-27$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$8x^{2}-40x-192-3x^{2}=-27$$

5x^{2} ni olish uchun 8x^{2} va -3x^{2} ni birlashtirish.

$$5x^{2}-40x-192=-27$$

27 ni ikki tarafga qo’shing.

$$5x^{2}-40x-192+27=0$$

-165 olish uchun -192 va 27'ni qo'shing.

$$5x^{2}-40x-165=0$$

Ikki tarafini 5 ga bo‘ling.

$$x^{2}-8x-33=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-33 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=1\left(-33\right)=-33$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -33-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-33 3,-11$$

Har bir juftlik yigʻindisini hisoblang.

$$1-33=-32 3-11=-8$$

Yechim – -8 yigʻindisini beruvchi juftlik.

$$a=-11 b=3$$

x^{2}-8x-33 ni \left(x^{2}-11x\right)+\left(3x-33\right) sifatida qaytadan yozish.

$$\left(x^{2}-11x\right)+\left(3x-33\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$x\left(x-11\right)+3\left(x-11\right)$$

Distributiv funktsiyasidan foydalangan holda x-11 umumiy terminini chiqaring.

$$\left(x-11\right)\left(x+3\right)$$

Tenglamani yechish uchun x-11=0 va x+3=0 ni yeching.

$$x=11 x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=11$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 8\left(x-3\right)\left(x+3\right) ga, x+3,x^{2}-9,8 ning eng kichik karralisiga ko‘paytiring.

$$\left(8x-24\right)\left(x-2\right)-8\times 30=3\left(x-3\right)\left(x+3\right)$$

8x-24 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x+48-8\times 30=3\left(x-3\right)\left(x+3\right)$$

-240 hosil qilish uchun -8 va 30 ni ko'paytirish.

$$8x^{2}-40x+48-240=3\left(x-3\right)\left(x+3\right)$$

-192 olish uchun 48 dan 240 ni ayirish.

$$8x^{2}-40x-192=3\left(x-3\right)\left(x+3\right)$$

3 ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-40x-192=\left(3x-9\right)\left(x+3\right)$$

3x-9 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x-192=3x^{2}-27$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$8x^{2}-40x-192-3x^{2}=-27$$

5x^{2} ni olish uchun 8x^{2} va -3x^{2} ni birlashtirish.

$$5x^{2}-40x-192=-27$$

27 ni ikki tarafga qo’shing.

$$5x^{2}-40x-192+27=0$$

-165 olish uchun -192 va 27'ni qo'shing.

$$5x^{2}-40x-165=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -40 ni b va -165 ni c bilan almashtiring.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 5\left(-165\right)}}{2\times 5}$$

-40 kvadratini chiqarish.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 5\left(-165\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600-20\left(-165\right)}}{2\times 5}$$

-20 ni -165 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600+3300}}{2\times 5}$$

1600 ni 3300 ga qo'shish.

$$x=\frac{-\left(-40\right)±\sqrt{4900}}{2\times 5}$$

4900 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-40\right)±70}{2\times 5}$$

-40 ning teskarisi 40 ga teng.

$$x=\frac{40±70}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{40±70}{10}$$

x=\frac{40±70}{10} tenglamasini yeching, bunda ± musbat. 40 ni 70 ga qo'shish.

$$x=\frac{110}{10}$$

110 ni 10 ga bo'lish.

$$x=11$$

x=\frac{40±70}{10} tenglamasini yeching, bunda ± manfiy. 40 dan 70 ni ayirish.

$$x=-\frac{30}{10}$$

-30 ni 10 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=11 x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=11$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 8\left(x-3\right)\left(x+3\right) ga, x+3,x^{2}-9,8 ning eng kichik karralisiga ko‘paytiring.

$$\left(8x-24\right)\left(x-2\right)-8\times 30=3\left(x-3\right)\left(x+3\right)$$

8x-24 ga x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x+48-8\times 30=3\left(x-3\right)\left(x+3\right)$$

-240 hosil qilish uchun -8 va 30 ni ko'paytirish.

$$8x^{2}-40x+48-240=3\left(x-3\right)\left(x+3\right)$$

-192 olish uchun 48 dan 240 ni ayirish.

$$8x^{2}-40x-192=3\left(x-3\right)\left(x+3\right)$$

3 ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-40x-192=\left(3x-9\right)\left(x+3\right)$$

3x-9 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$8x^{2}-40x-192=3x^{2}-27$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$8x^{2}-40x-192-3x^{2}=-27$$

5x^{2} ni olish uchun 8x^{2} va -3x^{2} ni birlashtirish.

$$5x^{2}-40x-192=-27$$

192 ni ikki tarafga qo’shing.

$$5x^{2}-40x=-27+192$$

165 olish uchun -27 va 192'ni qo'shing.

$$5x^{2}-40x=165$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-40x}{5}=\frac{165}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{40}{5}\right)x=\frac{165}{5}$$

-40 ni 5 ga bo'lish.

$$x^{2}-8x=\frac{165}{5}$$

165 ni 5 ga bo'lish.

$$x^{2}-8x=33$$

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

$$x^{2}-8x+\left(-4\right)^{2}=33+\left(-4\right)^{2}$$

-4 kvadratini chiqarish.

$$x^{2}-8x+16=33+16$$

33 ni 16 ga qo'shish.

$$x^{2}-8x+16=49$$

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

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-4=7 x-4=-7$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$x=11 x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=11$$