30/x^2+5x+6-x/x+2=2x+1/x+3 eching

$$\frac { 30 } { x ^ { 2 } + 5 x + 6 } - \frac { x } { x + 2 } = \frac { 2 x + 1 } { x + 3 }$$

x uchun yechish

$$x = -\frac{14}{3} = -4\frac{2}{3} \approx -4,666666667$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

x qiymati -3,-2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+2\right)\left(x+3\right) ga, x^{2}+5x+6,x+2,x+3 ning eng kichik karralisiga ko‘paytiring.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

x^{2}+3x teskarisini topish uchun har birining teskarisini toping.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

x+2 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Ikkala tarafdan 5x ni ayirish.

$$30-3x^{2}-3x-5x=2$$

-8x ni olish uchun -3x va -5x ni birlashtirish.

$$30-3x^{2}-8x=2$$

Ikkala tarafdan 2 ni ayirish.

$$30-3x^{2}-8x-2=0$$

28 olish uchun 30 dan 2 ni ayirish.

$$28-3x^{2}-8x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-3x^{2}-8x+28=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -3x^{2}+ax+bx+28 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-8 ab=-3\times 28=-84$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga 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.

$$a=6 b=-14$$

-3x^{2}-8x+28 ni \left(-3x^{2}+6x\right)+\left(-14x+28\right) sifatida qaytadan yozish.

$$\left(-3x^{2}+6x\right)+\left(-14x+28\right)$$

Birinchi guruhda 3x ni va ikkinchi guruhda 14 ni faktordan chiqaring.

$$3x\left(-x+2\right)+14\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(3x+14\right)$$

Tenglamani yechish uchun -x+2=0 va 3x+14=0 ni yeching.

$$x=2 x=-\frac{14}{3}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,-2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+2\right)\left(x+3\right) ga, x^{2}+5x+6,x+2,x+3 ning eng kichik karralisiga ko‘paytiring.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

x^{2}+3x teskarisini topish uchun har birining teskarisini toping.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

x+2 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Ikkala tarafdan 5x ni ayirish.

$$30-3x^{2}-3x-5x=2$$

-8x ni olish uchun -3x va -5x ni birlashtirish.

$$30-3x^{2}-8x=2$$

Ikkala tarafdan 2 ni ayirish.

$$30-3x^{2}-8x-2=0$$

28 olish uchun 30 dan 2 ni ayirish.

$$28-3x^{2}-8x=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-3x^{2}-8x+28=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -8 ni b va 28 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+12\times 28}}{2\left(-3\right)}$$

12 ni 28 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+336}}{2\left(-3\right)}$$

64 ni 336 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{400}}{2\left(-3\right)}$$

400 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±20}{2\left(-3\right)}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±20}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{8±20}{-6}$$

x=\frac{8±20}{-6} tenglamasini yeching, bunda ± musbat. 8 ni 20 ga qo'shish.

$$x=\frac{28}{-6}$$

\frac{28}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{14}{3}$$

x=\frac{8±20}{-6} tenglamasini yeching, bunda ± manfiy. 8 dan 20 ni ayirish.

$$x=-\frac{12}{-6}$$

-12 ni -6 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=-\frac{14}{3} x=2$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,-2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+2\right)\left(x+3\right) ga, x^{2}+5x+6,x+2,x+3 ning eng kichik karralisiga ko‘paytiring.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

x^{2}+3x teskarisini topish uchun har birining teskarisini toping.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

x+2 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Ikkala tarafdan 5x ni ayirish.

$$30-3x^{2}-3x-5x=2$$

-8x ni olish uchun -3x va -5x ni birlashtirish.

$$30-3x^{2}-8x=2$$

Ikkala tarafdan 30 ni ayirish.

$$-3x^{2}-8x=2-30$$

-28 olish uchun 2 dan 30 ni ayirish.

$$-3x^{2}-8x=-28$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-8x}{-3}=-\frac{28}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{8}{-3}\right)x=-\frac{28}{-3}$$

-8 ni -3 ga bo'lish.

$$x^{2}+\frac{8}{3}x=-\frac{28}{-3}$$

-28 ni -3 ga bo'lish.

$$x^{2}+\frac{8}{3}x=\frac{28}{3}$$

\frac{8}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{3} olish uchun. Keyin, \frac{4}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{28}{3}+\left(\frac{4}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{3} kvadratini chiqarish.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{28}{3}+\frac{16}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{28}{3} ni \frac{16}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{100}{9}$$

x^{2}+\frac{8}{3}x+\frac{16}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{100}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Qisqartirish.

$$x+\frac{4}{3}=\frac{10}{3} x+\frac{4}{3}=-\frac{10}{3}$$

Tenglamaning ikkala tarafidan \frac{4}{3} ni ayirish.

$$x=2 x=-\frac{14}{3}$$