5/x-2-3/x+6=4/x eching

$$\frac { 5 } { x - 2 } - \frac { 3 } { x + 6 } = \frac { 4 } { x }$$

x uchun yechish

$$x=-2$$

$$x=12$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -6,0,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-2\right)\left(x+6\right) ga, x-2,x+6,x ning eng kichik karralisiga ko‘paytiring.

$$x\left(x+6\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

x ga x+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(x^{2}+6x\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

x^{2}+6x ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+30x-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(x^{2}-2x\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(3x^{2}-6x\right)=\left(x-2\right)\left(x+6\right)\times 4$$

3x^{2}-6x teskarisini topish uchun har birining teskarisini toping.

$$5x^{2}+30x-3x^{2}+6x=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$2x^{2}+30x+6x=\left(x-2\right)\left(x+6\right)\times 4$$

36x ni olish uchun 30x va 6x ni birlashtirish.

$$2x^{2}+36x=\left(x-2\right)\left(x+6\right)\times 4$$

x-2 ga x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+36x=\left(x^{2}+4x-12\right)\times 4$$

x^{2}+4x-12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+36x=4x^{2}+16x-48$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$2x^{2}+36x-4x^{2}=16x-48$$

-2x^{2} ni olish uchun 2x^{2} va -4x^{2} ni birlashtirish.

$$-2x^{2}+36x=16x-48$$

Ikkala tarafdan 16x ni ayirish.

$$-2x^{2}+36x-16x=-48$$

20x ni olish uchun 36x va -16x ni birlashtirish.

$$-2x^{2}+20x=-48$$

48 ni ikki tarafga qo’shing.

$$-2x^{2}+20x+48=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 20 ni b va 48 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-2\right)\times 48}}{2\left(-2\right)}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-2\right)\times 48}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+8\times 48}}{2\left(-2\right)}$$

8 ni 48 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+384}}{2\left(-2\right)}$$

400 ni 384 ga qo'shish.

$$x=\frac{-20±\sqrt{784}}{2\left(-2\right)}$$

784 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±28}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-20±28}{-4}$$

x=\frac{-20±28}{-4} tenglamasini yeching, bunda ± musbat. -20 ni 28 ga qo'shish.

$$x=\frac{8}{-4}$$

8 ni -4 ga bo'lish.

$$x=-2$$

x=\frac{-20±28}{-4} tenglamasini yeching, bunda ± manfiy. -20 dan 28 ni ayirish.

$$x=-\frac{48}{-4}$$

-48 ni -4 ga bo'lish.

$$x=12$$

Tenglama yechildi.

$$x=-2 x=12$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -6,0,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-2\right)\left(x+6\right) ga, x-2,x+6,x ning eng kichik karralisiga ko‘paytiring.

$$x\left(x+6\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

x ga x+6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(x^{2}+6x\right)\times 5-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

x^{2}+6x ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+30x-x\left(x-2\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(x^{2}-2x\right)\times 3=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$5x^{2}+30x-\left(3x^{2}-6x\right)=\left(x-2\right)\left(x+6\right)\times 4$$

3x^{2}-6x teskarisini topish uchun har birining teskarisini toping.

$$5x^{2}+30x-3x^{2}+6x=\left(x-2\right)\left(x+6\right)\times 4$$

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

$$2x^{2}+30x+6x=\left(x-2\right)\left(x+6\right)\times 4$$

36x ni olish uchun 30x va 6x ni birlashtirish.

$$2x^{2}+36x=\left(x-2\right)\left(x+6\right)\times 4$$

x-2 ga x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+36x=\left(x^{2}+4x-12\right)\times 4$$

x^{2}+4x-12 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+36x=4x^{2}+16x-48$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$2x^{2}+36x-4x^{2}=16x-48$$

-2x^{2} ni olish uchun 2x^{2} va -4x^{2} ni birlashtirish.

$$-2x^{2}+36x=16x-48$$

Ikkala tarafdan 16x ni ayirish.

$$-2x^{2}+36x-16x=-48$$

20x ni olish uchun 36x va -16x ni birlashtirish.

$$-2x^{2}+20x=-48$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+20x}{-2}=-\frac{48}{-2}$$

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

$$x^{2}+\frac{20}{-2}x=-\frac{48}{-2}$$

20 ni -2 ga bo'lish.

$$x^{2}-10x=-\frac{48}{-2}$$

-48 ni -2 ga bo'lish.

$$x^{2}-10x=24$$

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

$$x^{2}-10x+\left(-5\right)^{2}=24+\left(-5\right)^{2}$$

-5 kvadratini chiqarish.

$$x^{2}-10x+25=24+25$$

24 ni 25 ga qo'shish.

$$x^{2}-10x+25=49$$

x^{2}-10x+25 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-5\right)^{2}=49$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{49}$$

Qisqartirish.

$$x-5=7 x-5=-7$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$x=12 x=-2$$