15/p+6p-5/p+2=1 eching

$$\frac { 15 } { p } + \frac { 6 p - 5 } { p + 2 } = 1$$

p uchun yechish

$$p=\frac{-4+\sqrt{134}i}{5}\approx -0,8+2,315167381i$$

$$p=\frac{-\sqrt{134}i-4}{5}\approx -0,8-2,315167381i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(p+2\right)\times 15+p\left(6p-5\right)=p\left(p+2\right)$$

p+2 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15p+30+p\left(6p-5\right)=p\left(p+2\right)$$

p ga 6p-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15p+30+6p^{2}-5p=p\left(p+2\right)$$

10p ni olish uchun 15p va -5p ni birlashtirish.

$$10p+30+6p^{2}=p\left(p+2\right)$$

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

$$10p+30+6p^{2}=p^{2}+2p$$

Ikkala tarafdan p^{2} ni ayirish.

$$10p+30+6p^{2}-p^{2}=2p$$

5p^{2} ni olish uchun 6p^{2} va -p^{2} ni birlashtirish.

$$10p+30+5p^{2}=2p$$

Ikkala tarafdan 2p ni ayirish.

$$10p+30+5p^{2}-2p=0$$

8p ni olish uchun 10p va -2p ni birlashtirish.

$$8p+30+5p^{2}=0$$

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.

$$5p^{2}+8p+30=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 8 ni b va 30 ni c bilan almashtiring.

$$p=\frac{-8±\sqrt{8^{2}-4\times 5\times 30}}{2\times 5}$$

8 kvadratini chiqarish.

$$p=\frac{-8±\sqrt{64-4\times 5\times 30}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$p=\frac{-8±\sqrt{64-20\times 30}}{2\times 5}$$

-20 ni 30 marotabaga ko'paytirish.

$$p=\frac{-8±\sqrt{64-600}}{2\times 5}$$

64 ni -600 ga qo'shish.

$$p=\frac{-8±\sqrt{-536}}{2\times 5}$$

-536 ning kvadrat ildizini chiqarish.

$$p=\frac{-8±2\sqrt{134}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$p=\frac{-8±2\sqrt{134}i}{10}$$

p=\frac{-8±2\sqrt{134}i}{10} tenglamasini yeching, bunda ± musbat. -8 ni 2i\sqrt{134} ga qo'shish.

$$p=\frac{-8+2\sqrt{134}i}{10}$$

-8+2i\sqrt{134} ni 10 ga bo'lish.

$$p=\frac{-4+\sqrt{134}i}{5}$$

p=\frac{-8±2\sqrt{134}i}{10} tenglamasini yeching, bunda ± manfiy. -8 dan 2i\sqrt{134} ni ayirish.

$$p=\frac{-2\sqrt{134}i-8}{10}$$

-8-2i\sqrt{134} ni 10 ga bo'lish.

$$p=\frac{-\sqrt{134}i-4}{5}$$

Tenglama yechildi.

$$p=\frac{-4+\sqrt{134}i}{5} p=\frac{-\sqrt{134}i-4}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(p+2\right)\times 15+p\left(6p-5\right)=p\left(p+2\right)$$

p+2 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15p+30+p\left(6p-5\right)=p\left(p+2\right)$$

p ga 6p-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15p+30+6p^{2}-5p=p\left(p+2\right)$$

10p ni olish uchun 15p va -5p ni birlashtirish.

$$10p+30+6p^{2}=p\left(p+2\right)$$

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

$$10p+30+6p^{2}=p^{2}+2p$$

Ikkala tarafdan p^{2} ni ayirish.

$$10p+30+6p^{2}-p^{2}=2p$$

5p^{2} ni olish uchun 6p^{2} va -p^{2} ni birlashtirish.

$$10p+30+5p^{2}=2p$$

Ikkala tarafdan 2p ni ayirish.

$$10p+30+5p^{2}-2p=0$$

8p ni olish uchun 10p va -2p ni birlashtirish.

$$8p+30+5p^{2}=0$$

Ikkala tarafdan 30 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$8p+5p^{2}=-30$$

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

$$5p^{2}+8p=-30$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5p^{2}+8p}{5}=-\frac{30}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$p^{2}+\frac{8}{5}p=-\frac{30}{5}$$

-30 ni 5 ga bo'lish.

$$p^{2}+\frac{8}{5}p=-6$$

\frac{8}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{5} olish uchun. Keyin, \frac{4}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$p^{2}+\frac{8}{5}p+\left(\frac{4}{5}\right)^{2}=-6+\left(\frac{4}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{5} kvadratini chiqarish.

$$p^{2}+\frac{8}{5}p+\frac{16}{25}=-6+\frac{16}{25}$$

-6 ni \frac{16}{25} ga qo'shish.

$$p^{2}+\frac{8}{5}p+\frac{16}{25}=-\frac{134}{25}$$

p^{2}+\frac{8}{5}p+\frac{16}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(p+\frac{4}{5}\right)^{2}=-\frac{134}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(p+\frac{4}{5}\right)^{2}}=\sqrt{-\frac{134}{25}}$$

Qisqartirish.

$$p+\frac{4}{5}=\frac{\sqrt{134}i}{5} p+\frac{4}{5}=-\frac{\sqrt{134}i}{5}$$

Tenglamaning ikkala tarafidan \frac{4}{5} ni ayirish.

$$p=\frac{-4+\sqrt{134}i}{5} p=\frac{-\sqrt{134}i-4}{5}$$