1/2-x-1=1/x-2-6-x/3x^2-12 eching

$$\frac{ 1 }{ 2-x } -1 = \frac{ 1 }{ x-2 } - \frac{ 6-x }{ 3 { x }^{ 2 } -12 }$$

x uchun yechish

$$x=-3$$

$$x=\frac{2}{3}\approx 0,666666667$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(x-2\right)\left(x+2\right) ga, 2-x,x-2,3x^{2}-12 ning eng kichik karralisiga ko‘paytiring.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

-3 hosil qilish uchun 3 va -1 ni ko'paytirish.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

-3 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

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

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

6 olish uchun -6 va 12'ni qo'shing.

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

6-x teskarisini topish uchun har birining teskarisini toping.

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

0 olish uchun 6 dan 6 ni ayirish.

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

4x ni olish uchun 3x va x ni birlashtirish.

$$6-3x-3x^{2}=4x$$

Ikkala tarafdan 4x ni ayirish.

$$6-3x-3x^{2}-4x=0$$

-7x ni olish uchun -3x va -4x ni birlashtirish.

$$6-7x-3x^{2}=0$$

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

$$-3x^{2}-7x+6=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -3x^{2}+ax+bx+6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-7 ab=-3\times 6=-18$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -18-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-18 2,-9 3,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-18=-17 2-9=-7 3-6=-3$$

Yechim – -7 yigʻindisini beruvchi juftlik.

$$a=2 b=-9$$

-3x^{2}-7x+6 ni \left(-3x^{2}+2x\right)+\left(-9x+6\right) sifatida qaytadan yozish.

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

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

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

Distributiv funktsiyasidan foydalangan holda 3x-2 umumiy terminini chiqaring.

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

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

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

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(x-2\right)\left(x+2\right) ga, 2-x,x-2,3x^{2}-12 ning eng kichik karralisiga ko‘paytiring.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

-3 hosil qilish uchun 3 va -1 ni ko'paytirish.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

-3 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

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

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

6 olish uchun -6 va 12'ni qo'shing.

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

6-x teskarisini topish uchun har birining teskarisini toping.

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

0 olish uchun 6 dan 6 ni ayirish.

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

4x ni olish uchun 3x va x ni birlashtirish.

$$6-3x-3x^{2}=4x$$

Ikkala tarafdan 4x ni ayirish.

$$6-3x-3x^{2}-4x=0$$

-7x ni olish uchun -3x va -4x ni birlashtirish.

$$6-7x-3x^{2}=0$$

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.

$$-3x^{2}-7x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -7 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 6}}{2\left(-3\right)}$$

12 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\left(-3\right)}$$

49 ni 72 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\left(-3\right)}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±11}{2\left(-3\right)}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±11}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{7±11}{-6}$$

x=\frac{7±11}{-6} tenglamasini yeching, bunda ± musbat. 7 ni 11 ga qo'shish.

$$x=\frac{18}{-6}$$

18 ni -6 ga bo'lish.

$$x=-3$$

x=\frac{7±11}{-6} tenglamasini yeching, bunda ± manfiy. 7 dan 11 ni ayirish.

$$x=-\frac{4}{-6}$$

\frac{-4}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(x-2\right)\left(x+2\right) ga, 2-x,x-2,3x^{2}-12 ning eng kichik karralisiga ko‘paytiring.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

-3 hosil qilish uchun 3 va -1 ni ko'paytirish.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

-3 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

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

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

6 olish uchun -6 va 12'ni qo'shing.

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

6-x teskarisini topish uchun har birining teskarisini toping.

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

0 olish uchun 6 dan 6 ni ayirish.

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

4x ni olish uchun 3x va x ni birlashtirish.

$$6-3x-3x^{2}=4x$$

Ikkala tarafdan 4x ni ayirish.

$$6-3x-3x^{2}-4x=0$$

-7x ni olish uchun -3x va -4x ni birlashtirish.

$$6-7x-3x^{2}=0$$

Ikkala tarafdan 6 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-7x-3x^{2}=-6$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-3x^{2}-7x=-6$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-7x}{-3}=-\frac{6}{-3}$$

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

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{6}{-3}$$

-7 ni -3 ga bo'lish.

$$x^{2}+\frac{7}{3}x=-\frac{6}{-3}$$

-6 ni -3 ga bo'lish.

$$x^{2}+\frac{7}{3}x=2$$

\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{6} olish uchun. Keyin, \frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{6} kvadratini chiqarish.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=2+\frac{49}{36}$$

2 ni \frac{49}{36} ga qo'shish.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{121}{36}$$

x^{2}+\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{121}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Qisqartirish.

$$x+\frac{7}{6}=\frac{11}{6} x+\frac{7}{6}=-\frac{11}{6}$$

Tenglamaning ikkala tarafidan \frac{7}{6} ni ayirish.

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