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

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

x uchun yechish

$$x = \frac{\sqrt{1345} + 41}{4} \approx 19,41856041$$

$$x = \frac{41 - \sqrt{1345}}{4} \approx 1,08143959$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 14x\left(x-3\right) ga, x,2\left(x-3\right),7 ning eng kichik karralisiga ko‘paytiring.

$$14x-42+7x\times 3=2x\left(x-3\right)$$

21 hosil qilish uchun 7 va 3 ni ko'paytirish.

$$14x-42+21x=2x\left(x-3\right)$$

35x ni olish uchun 14x va 21x ni birlashtirish.

$$35x-42=2x\left(x-3\right)$$

2x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$35x-42=2x^{2}-6x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$35x-42-2x^{2}=-6x$$

6x ni ikki tarafga qo’shing.

$$35x-42-2x^{2}+6x=0$$

41x ni olish uchun 35x va 6x ni birlashtirish.

$$41x-42-2x^{2}=0$$

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.

$$-2x^{2}+41x-42=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 41 ni b va -42 ni c bilan almashtiring.

$$x=\frac{-41±\sqrt{41^{2}-4\left(-2\right)\left(-42\right)}}{2\left(-2\right)}$$

41 kvadratini chiqarish.

$$x=\frac{-41±\sqrt{1681-4\left(-2\right)\left(-42\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-41±\sqrt{1681+8\left(-42\right)}}{2\left(-2\right)}$$

8 ni -42 marotabaga ko'paytirish.

$$x=\frac{-41±\sqrt{1681-336}}{2\left(-2\right)}$$

1681 ni -336 ga qo'shish.

$$x=\frac{-41±\sqrt{1345}}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-41±\sqrt{1345}}{-4}$$

x=\frac{-41±\sqrt{1345}}{-4} tenglamasini yeching, bunda ± musbat. -41 ni \sqrt{1345} ga qo'shish.

$$x=\frac{\sqrt{1345}-41}{-4}$$

-41+\sqrt{1345} ni -4 ga bo'lish.

$$x=\frac{41-\sqrt{1345}}{4}$$

x=\frac{-41±\sqrt{1345}}{-4} tenglamasini yeching, bunda ± manfiy. -41 dan \sqrt{1345} ni ayirish.

$$x=\frac{-\sqrt{1345}-41}{-4}$$

-41-\sqrt{1345} ni -4 ga bo'lish.

$$x=\frac{\sqrt{1345}+41}{4}$$

Tenglama yechildi.

$$x=\frac{41-\sqrt{1345}}{4} x=\frac{\sqrt{1345}+41}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 14x\left(x-3\right) ga, x,2\left(x-3\right),7 ning eng kichik karralisiga ko‘paytiring.

$$14x-42+7x\times 3=2x\left(x-3\right)$$

21 hosil qilish uchun 7 va 3 ni ko'paytirish.

$$14x-42+21x=2x\left(x-3\right)$$

35x ni olish uchun 14x va 21x ni birlashtirish.

$$35x-42=2x\left(x-3\right)$$

2x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$35x-42=2x^{2}-6x$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$35x-42-2x^{2}=-6x$$

6x ni ikki tarafga qo’shing.

$$35x-42-2x^{2}+6x=0$$

41x ni olish uchun 35x va 6x ni birlashtirish.

$$41x-42-2x^{2}=0$$

42 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$41x-2x^{2}=42$$

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

$$-2x^{2}+41x=42$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+41x}{-2}=\frac{42}{-2}$$

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{41}{-2}x=\frac{42}{-2}$$

41 ni -2 ga bo'lish.

$$x^{2}-\frac{41}{2}x=\frac{42}{-2}$$

42 ni -2 ga bo'lish.

$$x^{2}-\frac{41}{2}x=-21$$

-\frac{41}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{41}{4} olish uchun. Keyin, -\frac{41}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{41}{2}x+\left(-\frac{41}{4}\right)^{2}=-21+\left(-\frac{41}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{41}{4} kvadratini chiqarish.

$$x^{2}-\frac{41}{2}x+\frac{1681}{16}=-21+\frac{1681}{16}$$

-21 ni \frac{1681}{16} ga qo'shish.

$$x^{2}-\frac{41}{2}x+\frac{1681}{16}=\frac{1345}{16}$$

x^{2}-\frac{41}{2}x+\frac{1681}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{41}{4}\right)^{2}=\frac{1345}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{41}{4}\right)^{2}}=\sqrt{\frac{1345}{16}}$$

Qisqartirish.

$$x-\frac{41}{4}=\frac{\sqrt{1345}}{4} x-\frac{41}{4}=-\frac{\sqrt{1345}}{4}$$

\frac{41}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{1345}+41}{4} x=\frac{41-\sqrt{1345}}{4}$$