6(1/x+1/x+5)=1 eching

$$6( \frac{ 1 }{ x } + \frac{ 1 }{ x+5 } )=1$$

x uchun yechish

$$x=-3$$

$$x=10$$

Guruhlash orqali faktorlash qadamlari

Ikki tarafini 6 ga bo‘ling.

$$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}$$

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 6x\left(x+5\right) ga, x,x+5,6 ning eng kichik karralisiga ko‘paytiring.

$$6x+30+6x=x\left(x+5\right)$$

12x ni olish uchun 6x va 6x ni birlashtirish.

$$12x+30=x\left(x+5\right)$$

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

$$12x+30=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$12x+30-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$12x+30-x^{2}-5x=0$$

7x ni olish uchun 12x va -5x ni birlashtirish.

$$7x+30-x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-x^{2}+7x+30=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+30 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=7 ab=-30=-30$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -30-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,30 -2,15 -3,10 -5,6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=10 b=-3$$

-x^{2}+7x+30 ni \left(-x^{2}+10x\right)+\left(-3x+30\right) sifatida qaytadan yozish.

$$\left(-x^{2}+10x\right)+\left(-3x+30\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -3 ni faktordan chiqaring.

$$-x\left(x-10\right)-3\left(x-10\right)$$

Distributiv funktsiyasidan foydalangan holda x-10 umumiy terminini chiqaring.

$$\left(x-10\right)\left(-x-3\right)$$

Tenglamani yechish uchun x-10=0 va -x-3=0 ni yeching.

$$x=10 x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikki tarafini 6 ga bo‘ling.

$$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}$$

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 6x\left(x+5\right) ga, x,x+5,6 ning eng kichik karralisiga ko‘paytiring.

$$6x+30+6x=x\left(x+5\right)$$

12x ni olish uchun 6x va 6x ni birlashtirish.

$$12x+30=x\left(x+5\right)$$

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

$$12x+30=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$12x+30-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$12x+30-x^{2}-5x=0$$

7x ni olish uchun 12x va -5x ni birlashtirish.

$$7x+30-x^{2}=0$$

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.

$$-x^{2}+7x+30=0$$

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 30 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+4\times 30}}{2\left(-1\right)}$$

4 ni 30 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+120}}{2\left(-1\right)}$$

49 ni 120 ga qo'shish.

$$x=\frac{-7±\sqrt{169}}{2\left(-1\right)}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±13}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-7±13}{-2}$$

x=\frac{-7±13}{-2} tenglamasini yeching, bunda ± musbat. -7 ni 13 ga qo'shish.

$$x=\frac{6}{-2}$$

6 ni -2 ga bo'lish.

$$x=-3$$

x=\frac{-7±13}{-2} tenglamasini yeching, bunda ± manfiy. -7 dan 13 ni ayirish.

$$x=-\frac{20}{-2}$$

-20 ni -2 ga bo'lish.

$$x=10$$

Tenglama yechildi.

$$x=-3 x=10$$

Kvadratni yakunlash bo'yicha qadamlar

Ikki tarafini 6 ga bo‘ling.

$$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}$$

x qiymati -5,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 6x\left(x+5\right) ga, x,x+5,6 ning eng kichik karralisiga ko‘paytiring.

$$6x+30+6x=x\left(x+5\right)$$

12x ni olish uchun 6x va 6x ni birlashtirish.

$$12x+30=x\left(x+5\right)$$

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

$$12x+30=x^{2}+5x$$

Ikkala tarafdan x^{2} ni ayirish.

$$12x+30-x^{2}=5x$$

Ikkala tarafdan 5x ni ayirish.

$$12x+30-x^{2}-5x=0$$

7x ni olish uchun 12x va -5x ni birlashtirish.

$$7x+30-x^{2}=0$$

Ikkala tarafdan 30 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$7x-x^{2}=-30$$

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

$$-x^{2}+7x=-30$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+7x}{-1}=-\frac{30}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{-1}x=-\frac{30}{-1}$$

7 ni -1 ga bo'lish.

$$x^{2}-7x=-\frac{30}{-1}$$

-30 ni -1 ga bo'lish.

$$x^{2}-7x=30$$

-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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=30+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=30+\frac{49}{4}$$

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

$$x^{2}-7x+\frac{49}{4}=\frac{169}{4}$$

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.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{169}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{13}{2} x-\frac{7}{2}=-\frac{13}{2}$$

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

$$x=10 x=-3$$