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

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

x uchun yechish

$$x = \frac{4 \sqrt{3}}{3} \approx 2,309401077$$

$$x = -\frac{4 \sqrt{3}}{3} \approx -2,309401077$$

Kvadrat ildizini topish bo'yicha qadamlar

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x-3,x-2,x^{2}-x-6 ning eng kichik karralisiga ko‘paytiring.

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

x^{2}-4 ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}-20+\left(x-3\right)\left(x+2\right)\times 2=\left(x-2\right)x$$

x-3 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}-20+\left(x^{2}-x-6\right)\times 2=\left(x-2\right)x$$

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

$$5x^{2}-20+2x^{2}-2x-12=\left(x-2\right)x$$

7x^{2} ni olish uchun 5x^{2} va 2x^{2} ni birlashtirish.

$$7x^{2}-20-2x-12=\left(x-2\right)x$$

-32 olish uchun -20 dan 12 ni ayirish.

$$7x^{2}-32-2x=\left(x-2\right)x$$

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

$$7x^{2}-32-2x=x^{2}-2x$$

Ikkala tarafdan x^{2} ni ayirish.

$$7x^{2}-32-2x-x^{2}=-2x$$

6x^{2} ni olish uchun 7x^{2} va -x^{2} ni birlashtirish.

$$6x^{2}-32-2x=-2x$$

2x ni ikki tarafga qo’shing.

$$6x^{2}-32-2x+2x=0$$

0 ni olish uchun -2x va 2x ni birlashtirish.

$$6x^{2}-32=0$$

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

$$6x^{2}=32$$

Ikki tarafini 6 ga bo‘ling.

$$x^{2}=\frac{32}{6}$$

\frac{32}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}=\frac{16}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x-3,x-2,x^{2}-x-6 ning eng kichik karralisiga ko‘paytiring.

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

x^{2}-4 ga 5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}-20+\left(x-3\right)\left(x+2\right)\times 2=\left(x-2\right)x$$

x-3 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x^{2}-20+\left(x^{2}-x-6\right)\times 2=\left(x-2\right)x$$

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

$$5x^{2}-20+2x^{2}-2x-12=\left(x-2\right)x$$

7x^{2} ni olish uchun 5x^{2} va 2x^{2} ni birlashtirish.

$$7x^{2}-20-2x-12=\left(x-2\right)x$$

-32 olish uchun -20 dan 12 ni ayirish.

$$7x^{2}-32-2x=\left(x-2\right)x$$

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

$$7x^{2}-32-2x=x^{2}-2x$$

Ikkala tarafdan x^{2} ni ayirish.

$$7x^{2}-32-2x-x^{2}=-2x$$

6x^{2} ni olish uchun 7x^{2} va -x^{2} ni birlashtirish.

$$6x^{2}-32-2x=-2x$$

2x ni ikki tarafga qo’shing.

$$6x^{2}-32-2x+2x=0$$

0 ni olish uchun -2x va 2x ni birlashtirish.

$$6x^{2}-32=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, 0 ni b va -32 ni c bilan almashtiring.

$$x=\frac{0±\sqrt{0^{2}-4\times 6\left(-32\right)}}{2\times 6}$$

0 kvadratini chiqarish.

$$x=\frac{0±\sqrt{-4\times 6\left(-32\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{-24\left(-32\right)}}{2\times 6}$$

-24 ni -32 marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{768}}{2\times 6}$$

768 ning kvadrat ildizini chiqarish.

$$x=\frac{0±16\sqrt{3}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{0±16\sqrt{3}}{12}$$

x=\frac{0±16\sqrt{3}}{12} tenglamasini yeching, bunda ± musbat.

$$x=\frac{4\sqrt{3}}{3}$$

x=\frac{0±16\sqrt{3}}{12} tenglamasini yeching, bunda ± manfiy.

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

Tenglama yechildi.

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