3x/+4-5-x/x+1=2 eching

$$\frac { 3 x } { + 4 } - \frac { 5 - x } { x + 1 } = 2$$

x uchun yechish

$$x = \frac{\sqrt{337} + 1}{6} \approx 3,226259958$$

$$x=\frac{1-\sqrt{337}}{6}\approx -2,892926625$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+1\right) ga, 4,x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+1\right)\times 3x-4\left(5-x\right)=8\left(x+1\right)$$

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(3x+3\right)x-4\left(5-x\right)=8\left(x+1\right)$$

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

$$3x^{2}+3x-4\left(5-x\right)=8\left(x+1\right)$$

-4 ga 5-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+3x-20+4x=8\left(x+1\right)$$

7x ni olish uchun 3x va 4x ni birlashtirish.

$$3x^{2}+7x-20=8\left(x+1\right)$$

8 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+7x-20=8x+8$$

Ikkala tarafdan 8x ni ayirish.

$$3x^{2}+7x-20-8x=8$$

-x ni olish uchun 7x va -8x ni birlashtirish.

$$3x^{2}-x-20=8$$

Ikkala tarafdan 8 ni ayirish.

$$3x^{2}-x-20-8=0$$

-28 olish uchun -20 dan 8 ni ayirish.

$$3x^{2}-x-28=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -1 ni b va -28 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-28\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-12\left(-28\right)}}{2\times 3}$$

-12 ni -28 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+336}}{2\times 3}$$

1 ni 336 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{337}}{2\times 3}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{337}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{337}}{6}$$

x=\frac{1±\sqrt{337}}{6} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{337} ga qo'shish.

$$x=\frac{\sqrt{337}+1}{6}$$

x=\frac{1±\sqrt{337}}{6} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{337} ni ayirish.

$$x=\frac{1-\sqrt{337}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{337}+1}{6} x=\frac{1-\sqrt{337}}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+1\right) ga, 4,x+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+1\right)\times 3x-4\left(5-x\right)=8\left(x+1\right)$$

x+1 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(3x+3\right)x-4\left(5-x\right)=8\left(x+1\right)$$

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

$$3x^{2}+3x-4\left(5-x\right)=8\left(x+1\right)$$

-4 ga 5-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+3x-20+4x=8\left(x+1\right)$$

7x ni olish uchun 3x va 4x ni birlashtirish.

$$3x^{2}+7x-20=8\left(x+1\right)$$

8 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+7x-20=8x+8$$

Ikkala tarafdan 8x ni ayirish.

$$3x^{2}+7x-20-8x=8$$

-x ni olish uchun 7x va -8x ni birlashtirish.

$$3x^{2}-x-20=8$$

20 ni ikki tarafga qo’shing.

$$3x^{2}-x=8+20$$

28 olish uchun 8 va 20'ni qo'shing.

$$3x^{2}-x=28$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-x}{3}=\frac{28}{3}$$

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

$$x^{2}-\frac{1}{3}x=\frac{28}{3}$$

-\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{6} olish uchun. Keyin, -\frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=\frac{28}{3}+\left(-\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{6} kvadratini chiqarish.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{28}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{28}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{337}{36}$$

x^{2}-\frac{1}{3}x+\frac{1}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{337}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{337}{36}}$$

Qisqartirish.

$$x-\frac{1}{6}=\frac{\sqrt{337}}{6} x-\frac{1}{6}=-\frac{\sqrt{337}}{6}$$

\frac{1}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{337}+1}{6} x=\frac{1-\sqrt{337}}{6}$$