x-2/x-1-x-1/x-2=x^2/x^2-3x+2 eching

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

x uchun yechish

$$x=-3$$

Guruhlash orqali faktorlash qadamlari

x qiymati 1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right) ga, x-1,x-2,x^{2}-3x+2 ning eng kichik karralisiga ko‘paytiring.

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

\left(x-2\right)^{2} hosil qilish uchun x-2 va x-2 ni ko'paytirish.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x-1\right)^{2}=x^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x^{2}-2x+1\right)=x^{2}$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$x^{2}-4x+4-x^{2}+2x-1=x^{2}$$

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

$$-4x+4+2x-1=x^{2}$$

-2x ni olish uchun -4x va 2x ni birlashtirish.

$$-2x+4-1=x^{2}$$

3 olish uchun 4 dan 1 ni ayirish.

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

Ikkala tarafdan x^{2} ni ayirish.

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

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

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

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

$$a+b=-2 ab=-3=-3$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=1 b=-3$$

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

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

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

$$x\left(-x+1\right)+3\left(-x+1\right)$$

Distributiv funktsiyasidan foydalangan holda -x+1 umumiy terminini chiqaring.

$$\left(-x+1\right)\left(x+3\right)$$

Tenglamani yechish uchun -x+1=0 va x+3=0 ni yeching.

$$x=1 x=-3$$

x qiymati 1 teng bo‘lmaydi.

$$x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right) ga, x-1,x-2,x^{2}-3x+2 ning eng kichik karralisiga ko‘paytiring.

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

\left(x-2\right)^{2} hosil qilish uchun x-2 va x-2 ni ko'paytirish.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x-1\right)^{2}=x^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x^{2}-2x+1\right)=x^{2}$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$x^{2}-4x+4-x^{2}+2x-1=x^{2}$$

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

$$-4x+4+2x-1=x^{2}$$

-2x ni olish uchun -4x va 2x ni birlashtirish.

$$-2x+4-1=x^{2}$$

3 olish uchun 4 dan 1 ni ayirish.

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

Ikkala tarafdan x^{2} ni ayirish.

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

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -2 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 3}}{2\left(-1\right)}$$

4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+12}}{2\left(-1\right)}$$

4 ni 12 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{16}}{2\left(-1\right)}$$

16 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±4}{2\left(-1\right)}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±4}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{2±4}{-2}$$

x=\frac{2±4}{-2} tenglamasini yeching, bunda ± musbat. 2 ni 4 ga qo'shish.

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

6 ni -2 ga bo'lish.

$$x=-3$$

x=\frac{2±4}{-2} tenglamasini yeching, bunda ± manfiy. 2 dan 4 ni ayirish.

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

-2 ni -2 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=-3 x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=-3$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right) ga, x-1,x-2,x^{2}-3x+2 ning eng kichik karralisiga ko‘paytiring.

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

\left(x-2\right)^{2} hosil qilish uchun x-2 va x-2 ni ko'paytirish.

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

\left(x-1\right)^{2} hosil qilish uchun x-1 va x-1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-2\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x-1\right)^{2}=x^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-\left(x^{2}-2x+1\right)=x^{2}$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$x^{2}-4x+4-x^{2}+2x-1=x^{2}$$

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

$$-4x+4+2x-1=x^{2}$$

-2x ni olish uchun -4x va 2x ni birlashtirish.

$$-2x+4-1=x^{2}$$

3 olish uchun 4 dan 1 ni ayirish.

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

Ikkala tarafdan x^{2} ni ayirish.

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

Ikkala tarafdan 3 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

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

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

Ikki tarafini -1 ga bo‘ling.

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

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

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

-2 ni -1 ga bo'lish.

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

-3 ni -1 ga bo'lish.

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

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+2x+1^{2}=3+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=3+1$$

3 ni 1 ga qo'shish.

$$x^{2}+2x+1=4$$

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+1\right)^{2}=4$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{4}$$

Qisqartirish.

$$x+1=2 x+1=-2$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=1 x=-3$$

x qiymati 1 teng bo‘lmaydi.

$$x=-3$$