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

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

x uchun yechish

$$x = \frac{\sqrt{57} + 9}{2} \approx 8,274917218$$

$$x=\frac{9-\sqrt{57}}{2}\approx 0,725082782$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x\times 4+\left(x-3\right)\times 2=x\left(x-3\right)$$

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

$$x\times 4+2x-6=x\left(x-3\right)$$

6x ni olish uchun x\times 4 va 2x ni birlashtirish.

$$6x-6=x\left(x-3\right)$$

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

$$6x-6=x^{2}-3x$$

Ikkala tarafdan x^{2} ni ayirish.

$$6x-6-x^{2}=-3x$$

3x ni ikki tarafga qo’shing.

$$6x-6-x^{2}+3x=0$$

9x ni olish uchun 6x va 3x ni birlashtirish.

$$9x-6-x^{2}=0$$

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.

$$-x^{2}+9x-6=0$$

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 -6 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+4\left(-6\right)}}{2\left(-1\right)}$$

4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-24}}{2\left(-1\right)}$$

81 ni -24 ga qo'shish.

$$x=\frac{-9±\sqrt{57}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{57}}{-2}$$

x=\frac{-9±\sqrt{57}}{-2} tenglamasini yeching, bunda ± musbat. -9 ni \sqrt{57} ga qo'shish.

$$x=\frac{\sqrt{57}-9}{-2}$$

-9+\sqrt{57} ni -2 ga bo'lish.

$$x=\frac{9-\sqrt{57}}{2}$$

x=\frac{-9±\sqrt{57}}{-2} tenglamasini yeching, bunda ± manfiy. -9 dan \sqrt{57} ni ayirish.

$$x=\frac{-\sqrt{57}-9}{-2}$$

-9-\sqrt{57} ni -2 ga bo'lish.

$$x=\frac{\sqrt{57}+9}{2}$$

Tenglama yechildi.

$$x=\frac{9-\sqrt{57}}{2} x=\frac{\sqrt{57}+9}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x\times 4+\left(x-3\right)\times 2=x\left(x-3\right)$$

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

$$x\times 4+2x-6=x\left(x-3\right)$$

6x ni olish uchun x\times 4 va 2x ni birlashtirish.

$$6x-6=x\left(x-3\right)$$

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

$$6x-6=x^{2}-3x$$

Ikkala tarafdan x^{2} ni ayirish.

$$6x-6-x^{2}=-3x$$

3x ni ikki tarafga qo’shing.

$$6x-6-x^{2}+3x=0$$

9x ni olish uchun 6x va 3x ni birlashtirish.

$$9x-6-x^{2}=0$$

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

$$9x-x^{2}=6$$

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

$$-x^{2}+9x=6$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+9x}{-1}=\frac{6}{-1}$$

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

$$x^{2}+\frac{9}{-1}x=\frac{6}{-1}$$

9 ni -1 ga bo'lish.

$$x^{2}-9x=\frac{6}{-1}$$

6 ni -1 ga bo'lish.

$$x^{2}-9x=-6$$

-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.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-6+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=-6+\frac{81}{4}$$

-6 ni \frac{81}{4} ga qo'shish.

$$x^{2}-9x+\frac{81}{4}=\frac{57}{4}$$

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.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{57}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Qisqartirish.

$$x-\frac{9}{2}=\frac{\sqrt{57}}{2} x-\frac{9}{2}=-\frac{\sqrt{57}}{2}$$

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

$$x=\frac{\sqrt{57}+9}{2} x=\frac{9-\sqrt{57}}{2}$$