2(2x-1/x+3)-3(x+3/2x-1)=5 eching

$$2 ( \frac { 2 x - 1 } { x + 3 } ) - 3 ( \frac { x + 3 } { 2 x - 1 } ) = 5$$

x uchun yechish

$$x=-10$$

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

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$2\left(2x-1\right)\left(2x-1\right)-3\left(x+3\right)\left(x+3\right)=5\left(2x-1\right)\left(x+3\right)$$

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

$$2\left(2x-1\right)^{2}-3\left(x+3\right)\left(x+3\right)=5\left(2x-1\right)\left(x+3\right)$$

\left(x+3\right)^{2} hosil qilish uchun x+3 va x+3 ni ko'paytirish.

$$2\left(2x-1\right)^{2}-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

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

$$2\left(4x^{2}-4x+1\right)-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

2 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+3\right)^{2} kengaytirilishi uchun ishlating.

$$8x^{2}-8x+2-3\left(x^{2}+6x+9\right)=5\left(2x-1\right)\left(x+3\right)$$

-3 ga x^{2}+6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-3x^{2}-18x-27=5\left(2x-1\right)\left(x+3\right)$$

5x^{2} ni olish uchun 8x^{2} va -3x^{2} ni birlashtirish.

$$5x^{2}-8x+2-18x-27=5\left(2x-1\right)\left(x+3\right)$$

-26x ni olish uchun -8x va -18x ni birlashtirish.

$$5x^{2}-26x+2-27=5\left(2x-1\right)\left(x+3\right)$$

-25 olish uchun 2 dan 27 ni ayirish.

$$5x^{2}-26x-25=5\left(2x-1\right)\left(x+3\right)$$

5 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}-26x-25=\left(10x-5\right)\left(x+3\right)$$

10x-5 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}-26x-25=10x^{2}+25x-15$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$5x^{2}-26x-25-10x^{2}=25x-15$$

-5x^{2} ni olish uchun 5x^{2} va -10x^{2} ni birlashtirish.

$$-5x^{2}-26x-25=25x-15$$

Ikkala tarafdan 25x ni ayirish.

$$-5x^{2}-26x-25-25x=-15$$

-51x ni olish uchun -26x va -25x ni birlashtirish.

$$-5x^{2}-51x-25=-15$$

15 ni ikki tarafga qo’shing.

$$-5x^{2}-51x-25+15=0$$

-10 olish uchun -25 va 15'ni qo'shing.

$$-5x^{2}-51x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, -51 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\left(-5\right)\left(-10\right)}}{2\left(-5\right)}$$

-51 kvadratini chiqarish.

$$x=\frac{-\left(-51\right)±\sqrt{2601-4\left(-5\right)\left(-10\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-51\right)±\sqrt{2601+20\left(-10\right)}}{2\left(-5\right)}$$

20 ni -10 marotabaga ko'paytirish.

$$x=\frac{-\left(-51\right)±\sqrt{2601-200}}{2\left(-5\right)}$$

2601 ni -200 ga qo'shish.

$$x=\frac{-\left(-51\right)±\sqrt{2401}}{2\left(-5\right)}$$

2401 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-51\right)±49}{2\left(-5\right)}$$

-51 ning teskarisi 51 ga teng.

$$x=\frac{51±49}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{51±49}{-10}$$

x=\frac{51±49}{-10} tenglamasini yeching, bunda ± musbat. 51 ni 49 ga qo'shish.

$$x=\frac{100}{-10}$$

100 ni -10 ga bo'lish.

$$x=-10$$

x=\frac{51±49}{-10} tenglamasini yeching, bunda ± manfiy. 51 dan 49 ni ayirish.

$$x=\frac{2}{-10}$$

\frac{2}{-10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{1}{5}$$

Tenglama yechildi.

$$x=-10 x=-\frac{1}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$2\left(2x-1\right)\left(2x-1\right)-3\left(x+3\right)\left(x+3\right)=5\left(2x-1\right)\left(x+3\right)$$

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

$$2\left(2x-1\right)^{2}-3\left(x+3\right)\left(x+3\right)=5\left(2x-1\right)\left(x+3\right)$$

\left(x+3\right)^{2} hosil qilish uchun x+3 va x+3 ni ko'paytirish.

$$2\left(2x-1\right)^{2}-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

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

$$2\left(4x^{2}-4x+1\right)-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

2 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-3\left(x+3\right)^{2}=5\left(2x-1\right)\left(x+3\right)$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+3\right)^{2} kengaytirilishi uchun ishlating.

$$8x^{2}-8x+2-3\left(x^{2}+6x+9\right)=5\left(2x-1\right)\left(x+3\right)$$

-3 ga x^{2}+6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-8x+2-3x^{2}-18x-27=5\left(2x-1\right)\left(x+3\right)$$

5x^{2} ni olish uchun 8x^{2} va -3x^{2} ni birlashtirish.

$$5x^{2}-8x+2-18x-27=5\left(2x-1\right)\left(x+3\right)$$

-26x ni olish uchun -8x va -18x ni birlashtirish.

$$5x^{2}-26x+2-27=5\left(2x-1\right)\left(x+3\right)$$

-25 olish uchun 2 dan 27 ni ayirish.

$$5x^{2}-26x-25=5\left(2x-1\right)\left(x+3\right)$$

5 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}-26x-25=\left(10x-5\right)\left(x+3\right)$$

10x-5 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}-26x-25=10x^{2}+25x-15$$

Ikkala tarafdan 10x^{2} ni ayirish.

$$5x^{2}-26x-25-10x^{2}=25x-15$$

-5x^{2} ni olish uchun 5x^{2} va -10x^{2} ni birlashtirish.

$$-5x^{2}-26x-25=25x-15$$

Ikkala tarafdan 25x ni ayirish.

$$-5x^{2}-26x-25-25x=-15$$

-51x ni olish uchun -26x va -25x ni birlashtirish.

$$-5x^{2}-51x-25=-15$$

25 ni ikki tarafga qo’shing.

$$-5x^{2}-51x=-15+25$$

10 olish uchun -15 va 25'ni qo'shing.

$$-5x^{2}-51x=10$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}-51x}{-5}=\frac{10}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{51}{-5}\right)x=\frac{10}{-5}$$

-51 ni -5 ga bo'lish.

$$x^{2}+\frac{51}{5}x=\frac{10}{-5}$$

10 ni -5 ga bo'lish.

$$x^{2}+\frac{51}{5}x=-2$$

\frac{51}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{51}{10} olish uchun. Keyin, \frac{51}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{51}{5}x+\left(\frac{51}{10}\right)^{2}=-2+\left(\frac{51}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{51}{10} kvadratini chiqarish.

$$x^{2}+\frac{51}{5}x+\frac{2601}{100}=-2+\frac{2601}{100}$$

-2 ni \frac{2601}{100} ga qo'shish.

$$x^{2}+\frac{51}{5}x+\frac{2601}{100}=\frac{2401}{100}$$

x^{2}+\frac{51}{5}x+\frac{2601}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{51}{10}\right)^{2}=\frac{2401}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{51}{10}\right)^{2}}=\sqrt{\frac{2401}{100}}$$

Qisqartirish.

$$x+\frac{51}{10}=\frac{49}{10} x+\frac{51}{10}=-\frac{49}{10}$$

Tenglamaning ikkala tarafidan \frac{51}{10} ni ayirish.

$$x=-\frac{1}{5} x=-10$$