1/x+1+x+3/(x-2)(x+1)=7x/(x+1)(x-2)-x/(x+1) eching

$$\frac { 1 } { x + 1 } + \frac { x + 3 } { ( x - 2 ) ( x + 1 ) } = \frac { 7 x } { ( x + 1 ) ( x - 2 ) } - \frac { x } { ( x + 1 ) }$$

x uchun yechish

$$x = \frac{3 \sqrt{5} + 7}{2} \approx 6,854101966$$

$$x=\frac{7-3\sqrt{5}}{2}\approx 0,145898034$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right) ga, x+1,\left(x-2\right)\left(x+1\right) ning eng kichik karralisiga ko‘paytiring.

$$x-2+x+3=7x-\left(x-2\right)x$$

2x ni olish uchun x va x ni birlashtirish.

$$2x-2+3=7x-\left(x-2\right)x$$

1 olish uchun -2 va 3'ni qo'shing.

$$2x+1=7x-\left(x-2\right)x$$

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

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

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

$$2x+1=7x-x^{2}+2x$$

9x ni olish uchun 7x va 2x ni birlashtirish.

$$2x+1=9x-x^{2}$$

Ikkala tarafdan 9x ni ayirish.

$$2x+1-9x=-x^{2}$$

-7x ni olish uchun 2x va -9x ni birlashtirish.

$$-7x+1=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$-7x+1+x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$x^{2}-7x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -7 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4}}{2}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4}}{2}$$

49 ni -4 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{45}}{2}$$

45 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±3\sqrt{5}}{2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±3\sqrt{5}}{2}$$

x=\frac{7±3\sqrt{5}}{2} tenglamasini yeching, bunda ± musbat. 7 ni 3\sqrt{5} ga qo'shish.

$$x=\frac{3\sqrt{5}+7}{2}$$

x=\frac{7±3\sqrt{5}}{2} tenglamasini yeching, bunda ± manfiy. 7 dan 3\sqrt{5} ni ayirish.

$$x=\frac{7-3\sqrt{5}}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{5}+7}{2} x=\frac{7-3\sqrt{5}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x+1\right) ga, x+1,\left(x-2\right)\left(x+1\right) ning eng kichik karralisiga ko‘paytiring.

$$x-2+x+3=7x-\left(x-2\right)x$$

2x ni olish uchun x va x ni birlashtirish.

$$2x-2+3=7x-\left(x-2\right)x$$

1 olish uchun -2 va 3'ni qo'shing.

$$2x+1=7x-\left(x-2\right)x$$

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

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

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

$$2x+1=7x-x^{2}+2x$$

9x ni olish uchun 7x va 2x ni birlashtirish.

$$2x+1=9x-x^{2}$$

Ikkala tarafdan 9x ni ayirish.

$$2x+1-9x=-x^{2}$$

-7x ni olish uchun 2x va -9x ni birlashtirish.

$$-7x+1=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$-7x+1+x^{2}=0$$

Ikkala tarafdan 1 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-7x+x^{2}=-1$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$x^{2}-7x=-1$$

-7 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{2} olish uchun. Keyin, -\frac{7}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=-1+\frac{49}{4}$$

-1 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{45}{4}$$

x^{2}-7x+\frac{49}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{45}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{3\sqrt{5}}{2} x-\frac{7}{2}=-\frac{3\sqrt{5}}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{5}+7}{2} x=\frac{7-3\sqrt{5}}{2}$$