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

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

x uchun yechish

$$x=\frac{2\sqrt{6}}{3}-1\approx 0,632993162$$

$$x=-\frac{2\sqrt{6}}{3}-1\approx -2,632993162$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

x+2 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(3x+6\right)x=5$$

3x+6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+6x=5$$

Ikkala tarafdan 5 ni ayirish.

$$3x^{2}+6x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 6 ni b va -5 ni c bilan almashtiring.

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

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-5\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-12\left(-5\right)}}{2\times 3}$$

-12 ni -5 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+60}}{2\times 3}$$

36 ni 60 ga qo'shish.

$$x=\frac{-6±\sqrt{96}}{2\times 3}$$

96 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±4\sqrt{6}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-6±4\sqrt{6}}{6}$$

x=\frac{-6±4\sqrt{6}}{6} tenglamasini yeching, bunda ± musbat. -6 ni 4\sqrt{6} ga qo'shish.

$$x=\frac{4\sqrt{6}-6}{6}$$

-6+4\sqrt{6} ni 6 ga bo'lish.

$$x=\frac{2\sqrt{6}}{3}-1$$

x=\frac{-6±4\sqrt{6}}{6} tenglamasini yeching, bunda ± manfiy. -6 dan 4\sqrt{6} ni ayirish.

$$x=\frac{-4\sqrt{6}-6}{6}$$

-6-4\sqrt{6} ni 6 ga bo'lish.

$$x=-\frac{2\sqrt{6}}{3}-1$$

Tenglama yechildi.

$$x=\frac{2\sqrt{6}}{3}-1 x=-\frac{2\sqrt{6}}{3}-1$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

x+2 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(3x+6\right)x=5$$

3x+6 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+6x=5$$

Ikki tarafini 3 ga bo‘ling.

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

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

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

6 ni 3 ga bo'lish.

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

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+2x+1^{2}=\frac{5}{3}+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=\frac{5}{3}+1$$

\frac{5}{3} ni 1 ga qo'shish.

$$x^{2}+2x+1=\frac{8}{3}$$

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\frac{2\sqrt{6}}{3} x+1=-\frac{2\sqrt{6}}{3}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\frac{2\sqrt{6}}{3}-1 x=-\frac{2\sqrt{6}}{3}-1$$