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

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

x uchun yechish (complex solution)

$$x=\sqrt{2}-1\approx 0,414213562$$

$$x=-\left(\sqrt{2}+1\right)\approx -2,414213562$$

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.

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

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

$$\left(5x+10\right)x=5$$

5x+10 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+10x=5$$

Ikkala tarafdan 5 ni ayirish.

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

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

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

10 kvadratini chiqarish.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

-20 ni -5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100+100}}{2\times 5}$$

100 ni 100 ga qo'shish.

$$x=\frac{-10±\sqrt{200}}{2\times 5}$$

200 ning kvadrat ildizini chiqarish.

$$x=\frac{-10±10\sqrt{2}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±10\sqrt{2}}{10}$$

x=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± musbat. -10 ni 10\sqrt{2} ga qo'shish.

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

-10+10\sqrt{2} ni 10 ga bo'lish.

$$x=\sqrt{2}-1$$

x=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± manfiy. -10 dan 10\sqrt{2} ni ayirish.

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

-10-10\sqrt{2} ni 10 ga bo'lish.

$$x=-\sqrt{2}-1$$

Tenglama yechildi.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

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.

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

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

$$\left(5x+10\right)x=5$$

5x+10 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+10x=5$$

Ikki tarafini 5 ga bo‘ling.

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

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

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

10 ni 5 ga bo'lish.

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

5 ni 5 ga bo'lish.

$$x^{2}+2x=1$$

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.

$$x^{2}+2x+1^{2}=1+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=1+1$$

1 ni 1 ga qo'shish.

$$x^{2}+2x+1=2$$

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

x uchun yechish

$$x=\sqrt{2}-1\approx 0,414213562$$

$$x=-\sqrt{2}-1\approx -2,414213562$$

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.

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

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

$$\left(5x+10\right)x=5$$

5x+10 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+10x=5$$

Ikkala tarafdan 5 ni ayirish.

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

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

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

10 kvadratini chiqarish.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

-20 ni -5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100+100}}{2\times 5}$$

100 ni 100 ga qo'shish.

$$x=\frac{-10±\sqrt{200}}{2\times 5}$$

200 ning kvadrat ildizini chiqarish.

$$x=\frac{-10±10\sqrt{2}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±10\sqrt{2}}{10}$$

x=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± musbat. -10 ni 10\sqrt{2} ga qo'shish.

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

-10+10\sqrt{2} ni 10 ga bo'lish.

$$x=\sqrt{2}-1$$

x=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± manfiy. -10 dan 10\sqrt{2} ni ayirish.

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

-10-10\sqrt{2} ni 10 ga bo'lish.

$$x=-\sqrt{2}-1$$

Tenglama yechildi.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

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.

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

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

$$\left(5x+10\right)x=5$$

5x+10 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x^{2}+10x=5$$

Ikki tarafini 5 ga bo‘ling.

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

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

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

10 ni 5 ga bo'lish.

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

5 ni 5 ga bo'lish.

$$x^{2}+2x=1$$

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.

$$x^{2}+2x+1^{2}=1+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=1+1$$

1 ni 1 ga qo'shish.

$$x^{2}+2x+1=2$$

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$