1/1frac{7{9}x}+1/x+1/1frac{7{9}x+2}=1/12 eching

$$\frac { 1 } { 1 \frac { 7 } { 9 } x } + \frac { 1 } { x } + \frac { 1 } { 1 \frac { 7 } { 9 } x + 2 } = \frac { 1 } { 12 }$$

x uchun yechish

$$x = \frac{15 \sqrt{193} + 195}{16} \approx 25,21166624$$

$$x=\frac{195-15\sqrt{193}}{16}\approx -0,83666624$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 12x ga, x,12 ning eng kichik karralisiga ko‘paytiring.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

\frac{75}{4} olish uchun \frac{27}{4} va 12'ni qo'shing.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Ikkala tarafdan x ni ayirish.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Shartlarni qayta saralash.

$$-x+54\times \frac{1}{8x+9}x+\frac{75}{4}=0$$

x qiymati -\frac{9}{8} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(8x+9\right) ga, 8x+9,4 ning eng kichik karralisiga ko‘paytiring.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

-4 hosil qilish uchun -1 va 4 ni ko'paytirish.

$$-4x\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

-4x ga 8x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-32x^{2}-36x+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

216 hosil qilish uchun 54 va 4 ni ko'paytirish.

$$-32x^{2}-36x+216\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

216 hosil qilish uchun 216 va 1 ni ko'paytirish.

$$-32x^{2}-36x+216x+4\left(8x+9\right)\times \frac{75}{4}=0$$

180x ni olish uchun -36x va 216x ni birlashtirish.

$$-32x^{2}+180x+4\left(8x+9\right)\times \frac{75}{4}=0$$

75 hosil qilish uchun 4 va \frac{75}{4} ni ko'paytirish.

$$-32x^{2}+180x+75\left(8x+9\right)=0$$

75 ga 8x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-32x^{2}+180x+600x+675=0$$

780x ni olish uchun 180x va 600x ni birlashtirish.

$$-32x^{2}+780x+675=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -32 ni a, 780 ni b va 675 ni c bilan almashtiring.

$$x=\frac{-780±\sqrt{780^{2}-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

780 kvadratini chiqarish.

$$x=\frac{-780±\sqrt{608400-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

-4 ni -32 marotabaga ko'paytirish.

$$x=\frac{-780±\sqrt{608400+128\times 675}}{2\left(-32\right)}$$

128 ni 675 marotabaga ko'paytirish.

$$x=\frac{-780±\sqrt{608400+86400}}{2\left(-32\right)}$$

608400 ni 86400 ga qo'shish.

$$x=\frac{-780±\sqrt{694800}}{2\left(-32\right)}$$

694800 ning kvadrat ildizini chiqarish.

$$x=\frac{-780±60\sqrt{193}}{2\left(-32\right)}$$

2 ni -32 marotabaga ko'paytirish.

$$x=\frac{-780±60\sqrt{193}}{-64}$$

x=\frac{-780±60\sqrt{193}}{-64} tenglamasini yeching, bunda ± musbat. -780 ni 60\sqrt{193} ga qo'shish.

$$x=\frac{60\sqrt{193}-780}{-64}$$

-780+60\sqrt{193} ni -64 ga bo'lish.

$$x=\frac{195-15\sqrt{193}}{16}$$

x=\frac{-780±60\sqrt{193}}{-64} tenglamasini yeching, bunda ± manfiy. -780 dan 60\sqrt{193} ni ayirish.

$$x=\frac{-60\sqrt{193}-780}{-64}$$

-780-60\sqrt{193} ni -64 ga bo'lish.

$$x=\frac{15\sqrt{193}+195}{16}$$

Tenglama yechildi.

$$x=\frac{195-15\sqrt{193}}{16} x=\frac{15\sqrt{193}+195}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 12x ga, x,12 ning eng kichik karralisiga ko‘paytiring.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

\frac{75}{4} olish uchun \frac{27}{4} va 12'ni qo'shing.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Ikkala tarafdan x ni ayirish.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Ikkala tarafdan \frac{75}{4} ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$54x\left(8x+9\right)^{-1}-x=-\frac{75}{4}$$

Shartlarni qayta saralash.

$$-x+54\times \frac{1}{8x+9}x=-\frac{75}{4}$$

x qiymati -\frac{9}{8} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(8x+9\right) ga, 8x+9,4 ning eng kichik karralisiga ko‘paytiring.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

-4 hosil qilish uchun -1 va 4 ni ko'paytirish.

$$-4x\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

-4x ga 8x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-32x^{2}-36x+54\times 4\times 1x=-75\left(8x+9\right)$$

216 hosil qilish uchun 54 va 4 ni ko'paytirish.

$$-32x^{2}-36x+216\times 1x=-75\left(8x+9\right)$$

216 hosil qilish uchun 216 va 1 ni ko'paytirish.

$$-32x^{2}-36x+216x=-75\left(8x+9\right)$$

180x ni olish uchun -36x va 216x ni birlashtirish.

$$-32x^{2}+180x=-75\left(8x+9\right)$$

-75 ga 8x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-32x^{2}+180x=-600x-675$$

600x ni ikki tarafga qo’shing.

$$-32x^{2}+180x+600x=-675$$

780x ni olish uchun 180x va 600x ni birlashtirish.

$$-32x^{2}+780x=-675$$

Ikki tarafini -32 ga bo‘ling.

$$\frac{-32x^{2}+780x}{-32}=-\frac{675}{-32}$$

-32 ga bo'lish -32 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{780}{-32}x=-\frac{675}{-32}$$

\frac{780}{-32} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{195}{8}x=-\frac{675}{-32}$$

-675 ni -32 ga bo'lish.

$$x^{2}-\frac{195}{8}x=\frac{675}{32}$$

-\frac{195}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{195}{16} olish uchun. Keyin, -\frac{195}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{195}{8}x+\left(-\frac{195}{16}\right)^{2}=\frac{675}{32}+\left(-\frac{195}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{195}{16} kvadratini chiqarish.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{675}{32}+\frac{38025}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{675}{32} ni \frac{38025}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{43425}{256}$$

x^{2}-\frac{195}{8}x+\frac{38025}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{195}{16}\right)^{2}=\frac{43425}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{195}{16}\right)^{2}}=\sqrt{\frac{43425}{256}}$$

Qisqartirish.

$$x-\frac{195}{16}=\frac{15\sqrt{193}}{16} x-\frac{195}{16}=-\frac{15\sqrt{193}}{16}$$

\frac{195}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15\sqrt{193}+195}{16} x=\frac{195-15\sqrt{193}}{16}$$