4/x-3=5/2x+3 eching

$$\frac { 4 } { x } - 3 = \frac { 5 } { 2 x + 3 }$$

x uchun yechish

$$x=-2$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

x qiymati -\frac{3}{2},0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(2x+3\right) ga, x,2x+3 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+3\right)\times 4+x\left(2x+3\right)\left(-3\right)=x\times 5$$

2x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12+x\left(2x+3\right)\left(-3\right)=x\times 5$$

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

$$8x+12+\left(2x^{2}+3x\right)\left(-3\right)=x\times 5$$

2x^{2}+3x ga -3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12-6x^{2}-9x=x\times 5$$

-x ni olish uchun 8x va -9x ni birlashtirish.

$$-x+12-6x^{2}=x\times 5$$

Ikkala tarafdan x\times 5 ni ayirish.

$$-x+12-6x^{2}-x\times 5=0$$

-6x ni olish uchun -x va -x\times 5 ni birlashtirish.

$$-6x+12-6x^{2}=0$$

Ikki tarafini 6 ga bo‘ling.

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

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

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

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

$$a+b=-1 ab=-2=-2$$

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.

$$a=1 b=-2$$

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

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

Birinchi guruhda x ni va ikkinchi guruhda 2 ni faktordan chiqaring.

$$x\left(-x+1\right)+2\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+2\right)$$

Tenglamani yechish uchun -x+1=0 va x+2=0 ni yeching.

$$x=1 x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{3}{2},0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(2x+3\right) ga, x,2x+3 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+3\right)\times 4+x\left(2x+3\right)\left(-3\right)=x\times 5$$

2x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12+x\left(2x+3\right)\left(-3\right)=x\times 5$$

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

$$8x+12+\left(2x^{2}+3x\right)\left(-3\right)=x\times 5$$

2x^{2}+3x ga -3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12-6x^{2}-9x=x\times 5$$

-x ni olish uchun 8x va -9x ni birlashtirish.

$$-x+12-6x^{2}=x\times 5$$

Ikkala tarafdan x\times 5 ni ayirish.

$$-x+12-6x^{2}-x\times 5=0$$

-6x ni olish uchun -x va -x\times 5 ni birlashtirish.

$$-6x+12-6x^{2}=0$$

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.

$$-6x^{2}-6x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, -6 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-6\right)\times 12}}{2\left(-6\right)}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-6\right)\times 12}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+24\times 12}}{2\left(-6\right)}$$

24 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+288}}{2\left(-6\right)}$$

36 ni 288 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{324}}{2\left(-6\right)}$$

324 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±18}{2\left(-6\right)}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±18}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{6±18}{-12}$$

x=\frac{6±18}{-12} tenglamasini yeching, bunda ± musbat. 6 ni 18 ga qo'shish.

$$x=\frac{24}{-12}$$

24 ni -12 ga bo'lish.

$$x=-2$$

x=\frac{6±18}{-12} tenglamasini yeching, bunda ± manfiy. 6 dan 18 ni ayirish.

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

-12 ni -12 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=-2 x=1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{3}{2},0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(2x+3\right) ga, x,2x+3 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+3\right)\times 4+x\left(2x+3\right)\left(-3\right)=x\times 5$$

2x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12+x\left(2x+3\right)\left(-3\right)=x\times 5$$

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

$$8x+12+\left(2x^{2}+3x\right)\left(-3\right)=x\times 5$$

2x^{2}+3x ga -3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x+12-6x^{2}-9x=x\times 5$$

-x ni olish uchun 8x va -9x ni birlashtirish.

$$-x+12-6x^{2}=x\times 5$$

Ikkala tarafdan x\times 5 ni ayirish.

$$-x+12-6x^{2}-x\times 5=0$$

-6x ni olish uchun -x va -x\times 5 ni birlashtirish.

$$-6x+12-6x^{2}=0$$

Ikkala tarafdan 12 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-6x-6x^{2}=-12$$

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

$$-6x^{2}-6x=-12$$

Ikki tarafini -6 ga bo‘ling.

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

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

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

-6 ni -6 ga bo'lish.

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

-12 ni -6 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

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

2 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

x^{2}+x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

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

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=1 x=-2$$