x/x+3=6/x-3-27-x^2/9-x^2 eching

$$\frac { x } { x + 3 } = \frac { 6 } { x - 3 } - \frac { 27 - x ^ { 2 } } { 9 - x ^ { 2 } }$$

x uchun yechish

$$x = \frac{15}{2} = 7\frac{1}{2} = 7,5$$

Guruhlash orqali faktorlash qadamlari

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

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

x+3 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x=6x+18+27-x^{2}$$

45 olish uchun 18 va 27'ni qo'shing.

$$x^{2}-3x=6x+45-x^{2}$$

Ikkala tarafdan 6x ni ayirish.

$$x^{2}-3x-6x=45-x^{2}$$

-9x ni olish uchun -3x va -6x ni birlashtirish.

$$x^{2}-9x=45-x^{2}$$

Ikkala tarafdan 45 ni ayirish.

$$x^{2}-9x-45=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-9x-45+x^{2}=0$$

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

$$2x^{2}-9x-45=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-45 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-9 ab=2\left(-45\right)=-90$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -90-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Har bir juftlik yigʻindisini hisoblang.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Yechim – -9 yigʻindisini beruvchi juftlik.

$$a=-15 b=6$$

2x^{2}-9x-45 ni \left(2x^{2}-15x\right)+\left(6x-45\right) sifatida qaytadan yozish.

$$\left(2x^{2}-15x\right)+\left(6x-45\right)$$

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

$$x\left(2x-15\right)+3\left(2x-15\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-15 umumiy terminini chiqaring.

$$\left(2x-15\right)\left(x+3\right)$$

Tenglamani yechish uchun 2x-15=0 va x+3=0 ni yeching.

$$x=\frac{15}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{15}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

x+3 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x=6x+18+27-x^{2}$$

45 olish uchun 18 va 27'ni qo'shing.

$$x^{2}-3x=6x+45-x^{2}$$

Ikkala tarafdan 6x ni ayirish.

$$x^{2}-3x-6x=45-x^{2}$$

-9x ni olish uchun -3x va -6x ni birlashtirish.

$$x^{2}-9x=45-x^{2}$$

Ikkala tarafdan 45 ni ayirish.

$$x^{2}-9x-45=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-9x-45+x^{2}=0$$

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

$$2x^{2}-9x-45=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -9 ni b va -45 ni c bilan almashtiring.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-45\right)}}{2\times 2}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-45\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-45\right)}}{2\times 2}$$

-8 ni -45 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81+360}}{2\times 2}$$

81 ni 360 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{441}}{2\times 2}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-9\right)±21}{2\times 2}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±21}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{9±21}{4}$$

x=\frac{9±21}{4} tenglamasini yeching, bunda ± musbat. 9 ni 21 ga qo'shish.

$$x=\frac{30}{4}$$

\frac{30}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{15}{2}$$

x=\frac{9±21}{4} tenglamasini yeching, bunda ± manfiy. 9 dan 21 ni ayirish.

$$x=-\frac{12}{4}$$

-12 ni 4 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=\frac{15}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{15}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

x+3 ga 6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x=6x+18+27-x^{2}$$

45 olish uchun 18 va 27'ni qo'shing.

$$x^{2}-3x=6x+45-x^{2}$$

Ikkala tarafdan 6x ni ayirish.

$$x^{2}-3x-6x=45-x^{2}$$

-9x ni olish uchun -3x va -6x ni birlashtirish.

$$x^{2}-9x=45-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$x^{2}-9x+x^{2}=45$$

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

$$2x^{2}-9x=45$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-9x}{2}=\frac{45}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{9}{2}x=\frac{45}{2}$$

-\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{4} olish uchun. Keyin, -\frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=\frac{45}{2}+\left(-\frac{9}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{4} kvadratini chiqarish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{45}{2}+\frac{81}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{45}{2} ni \frac{81}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{441}{16}$$

x^{2}-\frac{9}{2}x+\frac{81}{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.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{441}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Qisqartirish.

$$x-\frac{9}{4}=\frac{21}{4} x-\frac{9}{4}=-\frac{21}{4}$$

\frac{9}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{15}{2}$$