2/b-3-6/2b+1=4 eching

$$\frac{2}{b-3}-\frac{6}{2b+1}=4$$

b uchun yechish

$$b = \frac{\sqrt{337} + 9}{8} \approx 3,419694969$$

$$b=\frac{9-\sqrt{337}}{8}\approx -1,169694969$$

Kvadrat tenglamalaridan foydalanish qoidalari

b qiymati -\frac{1}{2},3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(b-3\right)\left(2b+1\right) ga, b-3,2b+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

2b+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

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

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

6b-18 teskarisini topish uchun har birining teskarisini toping.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

-2b ni olish uchun 4b va -6b ni birlashtirish.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

20 olish uchun 2 va 18'ni qo'shing.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

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

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

4b-12 ga 2b+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-2b+20=8b^{2}-20b-12$$

Ikkala tarafdan 8b^{2} ni ayirish.

$$-2b+20-8b^{2}=-20b-12$$

20b ni ikki tarafga qo’shing.

$$-2b+20-8b^{2}+20b=-12$$

18b ni olish uchun -2b va 20b ni birlashtirish.

$$18b+20-8b^{2}=-12$$

12 ni ikki tarafga qo’shing.

$$18b+20-8b^{2}+12=0$$

32 olish uchun 20 va 12'ni qo'shing.

$$18b+32-8b^{2}=0$$

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.

$$-8b^{2}+18b+32=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 18 ni b va 32 ni c bilan almashtiring.

$$b=\frac{-18±\sqrt{18^{2}-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

18 kvadratini chiqarish.

$$b=\frac{-18±\sqrt{324-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$b=\frac{-18±\sqrt{324+32\times 32}}{2\left(-8\right)}$$

32 ni 32 marotabaga ko'paytirish.

$$b=\frac{-18±\sqrt{324+1024}}{2\left(-8\right)}$$

324 ni 1024 ga qo'shish.

$$b=\frac{-18±\sqrt{1348}}{2\left(-8\right)}$$

1348 ning kvadrat ildizini chiqarish.

$$b=\frac{-18±2\sqrt{337}}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$b=\frac{-18±2\sqrt{337}}{-16}$$

b=\frac{-18±2\sqrt{337}}{-16} tenglamasini yeching, bunda ± musbat. -18 ni 2\sqrt{337} ga qo'shish.

$$b=\frac{2\sqrt{337}-18}{-16}$$

-18+2\sqrt{337} ni -16 ga bo'lish.

$$b=\frac{9-\sqrt{337}}{8}$$

b=\frac{-18±2\sqrt{337}}{-16} tenglamasini yeching, bunda ± manfiy. -18 dan 2\sqrt{337} ni ayirish.

$$b=\frac{-2\sqrt{337}-18}{-16}$$

-18-2\sqrt{337} ni -16 ga bo'lish.

$$b=\frac{\sqrt{337}+9}{8}$$

Tenglama yechildi.

$$b=\frac{9-\sqrt{337}}{8} b=\frac{\sqrt{337}+9}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

b qiymati -\frac{1}{2},3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(b-3\right)\left(2b+1\right) ga, b-3,2b+1 ning eng kichik karralisiga ko‘paytiring.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

2b+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

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

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

6b-18 teskarisini topish uchun har birining teskarisini toping.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

-2b ni olish uchun 4b va -6b ni birlashtirish.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

20 olish uchun 2 va 18'ni qo'shing.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

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

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

4b-12 ga 2b+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-2b+20=8b^{2}-20b-12$$

Ikkala tarafdan 8b^{2} ni ayirish.

$$-2b+20-8b^{2}=-20b-12$$

20b ni ikki tarafga qo’shing.

$$-2b+20-8b^{2}+20b=-12$$

18b ni olish uchun -2b va 20b ni birlashtirish.

$$18b+20-8b^{2}=-12$$

Ikkala tarafdan 20 ni ayirish.

$$18b-8b^{2}=-12-20$$

-32 olish uchun -12 dan 20 ni ayirish.

$$18b-8b^{2}=-32$$

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

$$-8b^{2}+18b=-32$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8b^{2}+18b}{-8}=-\frac{32}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$b^{2}+\frac{18}{-8}b=-\frac{32}{-8}$$

\frac{18}{-8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$b^{2}-\frac{9}{4}b=-\frac{32}{-8}$$

-32 ni -8 ga bo'lish.

$$b^{2}-\frac{9}{4}b=4$$

-\frac{9}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{8} olish uchun. Keyin, -\frac{9}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$b^{2}-\frac{9}{4}b+\left(-\frac{9}{8}\right)^{2}=4+\left(-\frac{9}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{8} kvadratini chiqarish.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=4+\frac{81}{64}$$

4 ni \frac{81}{64} ga qo'shish.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=\frac{337}{64}$$

b^{2}-\frac{9}{4}b+\frac{81}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(b-\frac{9}{8}\right)^{2}=\frac{337}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(b-\frac{9}{8}\right)^{2}}=\sqrt{\frac{337}{64}}$$

Qisqartirish.

$$b-\frac{9}{8}=\frac{\sqrt{337}}{8} b-\frac{9}{8}=-\frac{\sqrt{337}}{8}$$

\frac{9}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$b=\frac{\sqrt{337}+9}{8} b=\frac{9-\sqrt{337}}{8}$$