3x-8/x-2=5x-2/x+5 eching

$$\frac { 3 x - 8 } { x - 2 } = \frac { 5 x - 2 } { x + 5 }$$

x uchun yechish

$$x=4$$

$$x = \frac{11}{2} = 5\frac{1}{2} = 5,5$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -5,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+5\right) ga, x-2,x+5 ning eng kichik karralisiga ko‘paytiring.

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

x+5 ga 3x-8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

x-2 ga 5x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+7x-40=5x^{2}-12x+4$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

12x ni ikki tarafga qo’shing.

$$-2x^{2}+7x-40+12x=4$$

19x ni olish uchun 7x va 12x ni birlashtirish.

$$-2x^{2}+19x-40=4$$

Ikkala tarafdan 4 ni ayirish.

$$-2x^{2}+19x-40-4=0$$

-44 olish uchun -40 dan 4 ni ayirish.

$$-2x^{2}+19x-44=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 19 ni b va -44 ni c bilan almashtiring.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

19 kvadratini chiqarish.

$$x=\frac{-19±\sqrt{361-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-19±\sqrt{361+8\left(-44\right)}}{2\left(-2\right)}$$

8 ni -44 marotabaga ko'paytirish.

$$x=\frac{-19±\sqrt{361-352}}{2\left(-2\right)}$$

361 ni -352 ga qo'shish.

$$x=\frac{-19±\sqrt{9}}{2\left(-2\right)}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-19±3}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-19±3}{-4}$$

x=\frac{-19±3}{-4} tenglamasini yeching, bunda ± musbat. -19 ni 3 ga qo'shish.

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

-16 ni -4 ga bo'lish.

$$x=4$$

x=\frac{-19±3}{-4} tenglamasini yeching, bunda ± manfiy. -19 dan 3 ni ayirish.

$$x=-\frac{22}{-4}$$

\frac{-22}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{11}{2}$$

Tenglama yechildi.

$$x=4 x=\frac{11}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -5,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+5\right) ga, x-2,x+5 ning eng kichik karralisiga ko‘paytiring.

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

x+5 ga 3x-8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

x-2 ga 5x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+7x-40=5x^{2}-12x+4$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

12x ni ikki tarafga qo’shing.

$$-2x^{2}+7x-40+12x=4$$

19x ni olish uchun 7x va 12x ni birlashtirish.

$$-2x^{2}+19x-40=4$$

40 ni ikki tarafga qo’shing.

$$-2x^{2}+19x=4+40$$

44 olish uchun 4 va 40'ni qo'shing.

$$-2x^{2}+19x=44$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+19x}{-2}=\frac{44}{-2}$$

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

$$x^{2}+\frac{19}{-2}x=\frac{44}{-2}$$

19 ni -2 ga bo'lish.

$$x^{2}-\frac{19}{2}x=\frac{44}{-2}$$

44 ni -2 ga bo'lish.

$$x^{2}-\frac{19}{2}x=-22$$

-\frac{19}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{4} olish uchun. Keyin, -\frac{19}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{2}x+\left(-\frac{19}{4}\right)^{2}=-22+\left(-\frac{19}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{4} kvadratini chiqarish.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=-22+\frac{361}{16}$$

-22 ni \frac{361}{16} ga qo'shish.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=\frac{9}{16}$$

x^{2}-\frac{19}{2}x+\frac{361}{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.

$$\left(x-\frac{19}{4}\right)^{2}=\frac{9}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Qisqartirish.

$$x-\frac{19}{4}=\frac{3}{4} x-\frac{19}{4}=-\frac{3}{4}$$

\frac{19}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{11}{2} x=4$$