3x+10/x-2[9x-4/3-3(x/2+7x-6/4)]=frac{7x+5}{2} eching

$$\frac { 3 x + 10 } { x } - 2 [ \frac { 9 x - 4 } { 3 } - 3 ( \frac { x } { 2 } + \frac { 7 x - 6 } { 4 } ) ] = \frac { 7 x + 5 } { 2 }$$

x uchun yechish (complex solution)

$$x=\frac{35+\sqrt{4535}i}{48}\approx 0,729166667+1,402966846i$$

$$x=\frac{-\sqrt{4535}i+35}{48}\approx 0,729166667-1,402966846i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 12x ga, x,3,2,4 ning eng kichik karralisiga ko‘paytiring.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

12 ga 3x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2 va 4 ning eng kichik umumiy karralisi 4. \frac{x}{2} ni \frac{2}{2} marotabaga ko'paytirish.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

\frac{2x}{4} va \frac{7x-6}{4} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

2x+7x-6 kabi iboralarga o‘xshab birlashtiring.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

3\times \frac{9x-6}{4} ni yagona kasrga aylantiring.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

3 ga 9x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3 va 4 ning eng kichik umumiy karralisi 12. \frac{9x-4}{3} ni \frac{4}{4} marotabaga ko'paytirish. \frac{27x-18}{4} ni \frac{3}{3} marotabaga ko'paytirish.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

\frac{4\left(9x-4\right)}{12} va \frac{3\left(27x-18\right)}{12} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

4\left(9x-4\right)-3\left(27x-18\right) ichidagi ko‘paytirishlarni bajaring.

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

36x-16-81x+54 kabi iboralarga o‘xshab birlashtiring.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

24 hosil qilish uchun 2 va 12 ni ko'paytirish.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

24 va 12 ichida eng katta umumiy 12 faktorini bekor qiling.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

6x ga 7x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Ikkala tarafdan 42x^{2} ni ayirish.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Ikkala tarafdan 30x ni ayirish.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

-2 ga -45x+38 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

90x-76 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

-40x ni olish uchun 36x va -76x ni birlashtirish.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

48x^{2} ni olish uchun 90x^{2} va -42x^{2} ni birlashtirish.

$$-40x+120+48x^{2}-30x=0$$

-70x ni olish uchun -40x va -30x ni birlashtirish.

$$-70x+120+48x^{2}=0$$

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.

$$48x^{2}-70x+120=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 48 ni a, -70 ni b va 120 ni c bilan almashtiring.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 48\times 120}}{2\times 48}$$

-70 kvadratini chiqarish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 48\times 120}}{2\times 48}$$

-4 ni 48 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-192\times 120}}{2\times 48}$$

-192 ni 120 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-23040}}{2\times 48}$$

4900 ni -23040 ga qo'shish.

$$x=\frac{-\left(-70\right)±\sqrt{-18140}}{2\times 48}$$

-18140 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-70\right)±2\sqrt{4535}i}{2\times 48}$$

-70 ning teskarisi 70 ga teng.

$$x=\frac{70±2\sqrt{4535}i}{2\times 48}$$

2 ni 48 marotabaga ko'paytirish.

$$x=\frac{70±2\sqrt{4535}i}{96}$$

x=\frac{70±2\sqrt{4535}i}{96} tenglamasini yeching, bunda ± musbat. 70 ni 2i\sqrt{4535} ga qo'shish.

$$x=\frac{70+2\sqrt{4535}i}{96}$$

70+2i\sqrt{4535} ni 96 ga bo'lish.

$$x=\frac{35+\sqrt{4535}i}{48}$$

x=\frac{70±2\sqrt{4535}i}{96} tenglamasini yeching, bunda ± manfiy. 70 dan 2i\sqrt{4535} ni ayirish.

$$x=\frac{-2\sqrt{4535}i+70}{96}$$

70-2i\sqrt{4535} ni 96 ga bo'lish.

$$x=\frac{-\sqrt{4535}i+35}{48}$$

Tenglama yechildi.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 12x ga, x,3,2,4 ning eng kichik karralisiga ko‘paytiring.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

12 ga 3x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2 va 4 ning eng kichik umumiy karralisi 4. \frac{x}{2} ni \frac{2}{2} marotabaga ko'paytirish.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

\frac{2x}{4} va \frac{7x-6}{4} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

2x+7x-6 kabi iboralarga o‘xshab birlashtiring.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

3\times \frac{9x-6}{4} ni yagona kasrga aylantiring.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

3 ga 9x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3 va 4 ning eng kichik umumiy karralisi 12. \frac{9x-4}{3} ni \frac{4}{4} marotabaga ko'paytirish. \frac{27x-18}{4} ni \frac{3}{3} marotabaga ko'paytirish.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

\frac{4\left(9x-4\right)}{12} va \frac{3\left(27x-18\right)}{12} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

4\left(9x-4\right)-3\left(27x-18\right) ichidagi ko‘paytirishlarni bajaring.

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

36x-16-81x+54 kabi iboralarga o‘xshab birlashtiring.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

24 hosil qilish uchun 2 va 12 ni ko'paytirish.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

24 va 12 ichida eng katta umumiy 12 faktorini bekor qiling.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

6x ga 7x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Ikkala tarafdan 42x^{2} ni ayirish.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Ikkala tarafdan 30x ni ayirish.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

-2 ga -45x+38 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

90x-76 ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

-40x ni olish uchun 36x va -76x ni birlashtirish.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

48x^{2} ni olish uchun 90x^{2} va -42x^{2} ni birlashtirish.

$$-40x+120+48x^{2}-30x=0$$

-70x ni olish uchun -40x va -30x ni birlashtirish.

$$-70x+120+48x^{2}=0$$

Ikkala tarafdan 120 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-70x+48x^{2}=-120$$

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

$$48x^{2}-70x=-120$$

Ikki tarafini 48 ga bo‘ling.

$$\frac{48x^{2}-70x}{48}=-\frac{120}{48}$$

48 ga bo'lish 48 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{70}{48}\right)x=-\frac{120}{48}$$

\frac{-70}{48} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{35}{24}x=-\frac{120}{48}$$

\frac{-120}{48} ulushini 24 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{35}{24}x=-\frac{5}{2}$$

-\frac{35}{24} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{35}{48} olish uchun. Keyin, -\frac{35}{48} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{35}{24}x+\left(-\frac{35}{48}\right)^{2}=-\frac{5}{2}+\left(-\frac{35}{48}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{35}{48} kvadratini chiqarish.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{5}{2}+\frac{1225}{2304}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{2} ni \frac{1225}{2304} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{4535}{2304}$$

x^{2}-\frac{35}{24}x+\frac{1225}{2304} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{35}{48}\right)^{2}=-\frac{4535}{2304}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{35}{48}\right)^{2}}=\sqrt{-\frac{4535}{2304}}$$

Qisqartirish.

$$x-\frac{35}{48}=\frac{\sqrt{4535}i}{48} x-\frac{35}{48}=-\frac{\sqrt{4535}i}{48}$$

\frac{35}{48} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$