3-x/left(x+1right)left(x+2right)=15/1 eching

$$\frac{ 3-x }{ \left( x+1 \right) \left( x+2 \right) } = \frac{ 15 }{ 1 }$$

x uchun yechish

$$x=\frac{2\sqrt{31}-23}{15}\approx -0,790964752$$

$$x=\frac{-2\sqrt{31}-23}{15}\approx -2,275701915$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(x+2\right) ga ko'paytirish.

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

x+1 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

x^{2}+3x+2 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3-x=15x^{2}+45x+30$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$3-x-15x^{2}=45x+30$$

Ikkala tarafdan 45x ni ayirish.

$$3-x-15x^{2}-45x=30$$

-46x ni olish uchun -x va -45x ni birlashtirish.

$$3-46x-15x^{2}=30$$

Ikkala tarafdan 30 ni ayirish.

$$3-46x-15x^{2}-30=0$$

-27 olish uchun 3 dan 30 ni ayirish.

$$-27-46x-15x^{2}=0$$

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.

$$-15x^{2}-46x-27=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -15 ni a, -46 ni b va -27 ni c bilan almashtiring.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

-46 kvadratini chiqarish.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-\left(-46\right)±\sqrt{2116+60\left(-27\right)}}{2\left(-15\right)}$$

60 ni -27 marotabaga ko'paytirish.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1620}}{2\left(-15\right)}$$

2116 ni -1620 ga qo'shish.

$$x=\frac{-\left(-46\right)±\sqrt{496}}{2\left(-15\right)}$$

496 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-46\right)±4\sqrt{31}}{2\left(-15\right)}$$

-46 ning teskarisi 46 ga teng.

$$x=\frac{46±4\sqrt{31}}{2\left(-15\right)}$$

2 ni -15 marotabaga ko'paytirish.

$$x=\frac{46±4\sqrt{31}}{-30}$$

x=\frac{46±4\sqrt{31}}{-30} tenglamasini yeching, bunda ± musbat. 46 ni 4\sqrt{31} ga qo'shish.

$$x=\frac{4\sqrt{31}+46}{-30}$$

46+4\sqrt{31} ni -30 ga bo'lish.

$$x=\frac{-2\sqrt{31}-23}{15}$$

x=\frac{46±4\sqrt{31}}{-30} tenglamasini yeching, bunda ± manfiy. 46 dan 4\sqrt{31} ni ayirish.

$$x=\frac{46-4\sqrt{31}}{-30}$$

46-4\sqrt{31} ni -30 ga bo'lish.

$$x=\frac{2\sqrt{31}-23}{15}$$

Tenglama yechildi.

$$x=\frac{-2\sqrt{31}-23}{15} x=\frac{2\sqrt{31}-23}{15}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(x+2\right) ga ko'paytirish.

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

x+1 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

x^{2}+3x+2 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3-x=15x^{2}+45x+30$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$3-x-15x^{2}=45x+30$$

Ikkala tarafdan 45x ni ayirish.

$$3-x-15x^{2}-45x=30$$

-46x ni olish uchun -x va -45x ni birlashtirish.

$$3-46x-15x^{2}=30$$

Ikkala tarafdan 3 ni ayirish.

$$-46x-15x^{2}=30-3$$

27 olish uchun 30 dan 3 ni ayirish.

$$-46x-15x^{2}=27$$

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

$$-15x^{2}-46x=27$$

Ikki tarafini -15 ga bo‘ling.

$$\frac{-15x^{2}-46x}{-15}=\frac{27}{-15}$$

-15 ga bo'lish -15 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{46}{-15}\right)x=\frac{27}{-15}$$

-46 ni -15 ga bo'lish.

$$x^{2}+\frac{46}{15}x=\frac{27}{-15}$$

\frac{27}{-15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{46}{15}x=-\frac{9}{5}$$

\frac{46}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{23}{15} olish uchun. Keyin, \frac{23}{15} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{46}{15}x+\left(\frac{23}{15}\right)^{2}=-\frac{9}{5}+\left(\frac{23}{15}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{23}{15} kvadratini chiqarish.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=-\frac{9}{5}+\frac{529}{225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{5} ni \frac{529}{225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=\frac{124}{225}$$

x^{2}+\frac{46}{15}x+\frac{529}{225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{23}{15}\right)^{2}=\frac{124}{225}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{23}{15}\right)^{2}}=\sqrt{\frac{124}{225}}$$

Qisqartirish.

$$x+\frac{23}{15}=\frac{2\sqrt{31}}{15} x+\frac{23}{15}=-\frac{2\sqrt{31}}{15}$$

Tenglamaning ikkala tarafidan \frac{23}{15} ni ayirish.

$$x=\frac{2\sqrt{31}-23}{15} x=\frac{-2\sqrt{31}-23}{15}$$