quad2/x=2-3/x-2 eching

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

x uchun yechish

$$x=4$$

$$x=\frac{1}{2}=0,5$$

Guruhlash orqali faktorlash qadamlari

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

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

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

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

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

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

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

$$2x-4=2x^{2}-4x-x\times 3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$2x-4-2x^{2}=-4x-x\times 3$$

4x ni ikki tarafga qo’shing.

$$2x-4-2x^{2}+4x=-x\times 3$$

6x ni olish uchun 2x va 4x ni birlashtirish.

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

x\times 3 ni ikki tarafga qo’shing.

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

9x ni olish uchun 6x va x\times 3 ni birlashtirish.

$$9x-4-2x^{2}=0$$

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

$$-2x^{2}+9x-4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=9 ab=-2\left(-4\right)=8$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 8-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,8 2,4$$

Har bir juftlik yigʻindisini hisoblang.

$$1+8=9 2+4=6$$

Yechim – 9 yigʻindisini beruvchi juftlik.

$$a=8 b=1$$

-2x^{2}+9x-4 ni \left(-2x^{2}+8x\right)+\left(x-4\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+8x\right)+\left(x-4\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$2x\left(-x+4\right)-\left(-x+4\right)$$

Distributiv funktsiyasidan foydalangan holda -x+4 umumiy terminini chiqaring.

$$\left(-x+4\right)\left(2x-1\right)$$

Tenglamani yechish uchun -x+4=0 va 2x-1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

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

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

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

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

$$2x-4=2x^{2}-4x-x\times 3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$2x-4-2x^{2}=-4x-x\times 3$$

4x ni ikki tarafga qo’shing.

$$2x-4-2x^{2}+4x=-x\times 3$$

6x ni olish uchun 2x va 4x ni birlashtirish.

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

x\times 3 ni ikki tarafga qo’shing.

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

9x ni olish uchun 6x va x\times 3 ni birlashtirish.

$$9x-4-2x^{2}=0$$

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.

$$-2x^{2}+9x-4=0$$

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 -4 ni c bilan almashtiring.

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

9 kvadratini chiqarish.

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

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+8\left(-4\right)}}{2\left(-2\right)}$$

8 ni -4 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-2\right)}$$

81 ni -32 ga qo'shish.

$$x=\frac{-9±\sqrt{49}}{2\left(-2\right)}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-9±7}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-9±7}{-4}$$

x=\frac{-9±7}{-4} tenglamasini yeching, bunda ± musbat. -9 ni 7 ga qo'shish.

$$x=-\frac{2}{-4}$$

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

$$x=\frac{1}{2}$$

x=\frac{-9±7}{-4} tenglamasini yeching, bunda ± manfiy. -9 dan 7 ni ayirish.

$$x=-\frac{16}{-4}$$

-16 ni -4 ga bo'lish.

$$x=4$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

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

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

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

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

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

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

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

$$2x-4=2x^{2}-4x-x\times 3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$2x-4-2x^{2}=-4x-x\times 3$$

4x ni ikki tarafga qo’shing.

$$2x-4-2x^{2}+4x=-x\times 3$$

6x ni olish uchun 2x va 4x ni birlashtirish.

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

x\times 3 ni ikki tarafga qo’shing.

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

9x ni olish uchun 6x va x\times 3 ni birlashtirish.

$$9x-4-2x^{2}=0$$

4 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$9x-2x^{2}=4$$

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

$$-2x^{2}+9x=4$$

Ikki tarafini -2 ga bo‘ling.

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

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

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

9 ni -2 ga bo'lish.

$$x^{2}-\frac{9}{2}x=\frac{4}{-2}$$

4 ni -2 ga bo'lish.

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

-\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.

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

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

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

-2 ni \frac{81}{16} ga qo'shish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

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.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x-\frac{9}{4}=\frac{7}{4} x-\frac{9}{4}=-\frac{7}{4}$$

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

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