1/x+4/x+1+1=15/x eching

$$\frac { 1 } { x } + \frac { 4 } { x + 1 } + 1 = \frac { 15 } { x }$$

x uchun yechish

$$x = \frac{\sqrt{137} + 9}{2} \approx 10,352349955$$

$$x=\frac{9-\sqrt{137}}{2}\approx -1,352349955$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+1\right) ga, x,x+1 ning eng kichik karralisiga ko‘paytiring.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

5x ni olish uchun x va x\times 4 ni birlashtirish.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

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

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

6x ni olish uchun 5x va x ni birlashtirish.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

x+1 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+1+x^{2}=15x+15$$

Ikkala tarafdan 15x ni ayirish.

$$6x+1+x^{2}-15x=15$$

-9x ni olish uchun 6x va -15x ni birlashtirish.

$$-9x+1+x^{2}=15$$

Ikkala tarafdan 15 ni ayirish.

$$-9x+1+x^{2}-15=0$$

-14 olish uchun 1 dan 15 ni ayirish.

$$-9x-14+x^{2}=0$$

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.

$$x^{2}-9x-14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -9 ni b va -14 ni c bilan almashtiring.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-14\right)}}{2}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-14\right)}}{2}$$

-4 ni -14 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81+56}}{2}$$

81 ni 56 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{137}}{2}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±\sqrt{137}}{2}$$

x=\frac{9±\sqrt{137}}{2} tenglamasini yeching, bunda ± musbat. 9 ni \sqrt{137} ga qo'shish.

$$x=\frac{\sqrt{137}+9}{2}$$

x=\frac{9±\sqrt{137}}{2} tenglamasini yeching, bunda ± manfiy. 9 dan \sqrt{137} ni ayirish.

$$x=\frac{9-\sqrt{137}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+1\right) ga, x,x+1 ning eng kichik karralisiga ko‘paytiring.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

5x ni olish uchun x va x\times 4 ni birlashtirish.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

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

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

6x ni olish uchun 5x va x ni birlashtirish.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

x+1 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+1+x^{2}=15x+15$$

Ikkala tarafdan 15x ni ayirish.

$$6x+1+x^{2}-15x=15$$

-9x ni olish uchun 6x va -15x ni birlashtirish.

$$-9x+1+x^{2}=15$$

Ikkala tarafdan 1 ni ayirish.

$$-9x+x^{2}=15-1$$

14 olish uchun 15 dan 1 ni ayirish.

$$-9x+x^{2}=14$$

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

$$x^{2}-9x=14$$

-9 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{2} olish uchun. Keyin, -\frac{9}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=14+\left(-\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x^{2}-9x+\frac{81}{4}=14+\frac{81}{4}$$

14 ni \frac{81}{4} ga qo'shish.

$$x^{2}-9x+\frac{81}{4}=\frac{137}{4}$$

x^{2}-9x+\frac{81}{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.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{137}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{137}{4}}$$

Qisqartirish.

$$x-\frac{9}{2}=\frac{\sqrt{137}}{2} x-\frac{9}{2}=-\frac{\sqrt{137}}{2}$$

\frac{9}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$