4/x^2-1+15+7x/x^4-5x^2+4=2/4-x^2 eching

$$\frac { 4 } { x ^ { 2 } - 1 } + \frac { 15 + 7 x } { x ^ { 4 } - 5 x ^ { 2 } + 4 } = \frac { 2 } { 4 - x ^ { 2 } }$$

x uchun yechish

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1,5$$

$$x=\frac{1}{3}\approx 0,333333333$$

Guruhlash orqali faktorlash qadamlari

x qiymati -2,-1,1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right) ga, x^{2}-1,x^{4}-5x^{2}+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-4\right)\times 4+15+7x=\left(-x^{2}+1\right)\times 2$$

x^{2}-4 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-16+15+7x=\left(-x^{2}+1\right)\times 2$$

-1 olish uchun -16 va 15'ni qo'shing.

$$4x^{2}-1+7x=\left(-x^{2}+1\right)\times 2$$

-x^{2}+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-1+7x=-2x^{2}+2$$

2x^{2} ni ikki tarafga qo’shing.

$$4x^{2}-1+7x+2x^{2}=2$$

6x^{2} ni olish uchun 4x^{2} va 2x^{2} ni birlashtirish.

$$6x^{2}-1+7x=2$$

Ikkala tarafdan 2 ni ayirish.

$$6x^{2}-1+7x-2=0$$

-3 olish uchun -1 dan 2 ni ayirish.

$$6x^{2}-3+7x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$6x^{2}+7x-3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 6x^{2}+ax+bx-3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=7 ab=6\left(-3\right)=-18$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -18-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,18 -2,9 -3,6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+18=17 -2+9=7 -3+6=3$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=-2 b=9$$

6x^{2}+7x-3 ni \left(6x^{2}-2x\right)+\left(9x-3\right) sifatida qaytadan yozish.

$$\left(6x^{2}-2x\right)+\left(9x-3\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

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

Distributiv funktsiyasidan foydalangan holda 3x-1 umumiy terminini chiqaring.

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

Tenglamani yechish uchun 3x-1=0 va 2x+3=0 ni yeching.

$$x=\frac{1}{3} x=-\frac{3}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,-1,1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right) ga, x^{2}-1,x^{4}-5x^{2}+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-4\right)\times 4+15+7x=\left(-x^{2}+1\right)\times 2$$

x^{2}-4 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-16+15+7x=\left(-x^{2}+1\right)\times 2$$

-1 olish uchun -16 va 15'ni qo'shing.

$$4x^{2}-1+7x=\left(-x^{2}+1\right)\times 2$$

-x^{2}+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-1+7x=-2x^{2}+2$$

2x^{2} ni ikki tarafga qo’shing.

$$4x^{2}-1+7x+2x^{2}=2$$

6x^{2} ni olish uchun 4x^{2} va 2x^{2} ni birlashtirish.

$$6x^{2}-1+7x=2$$

Ikkala tarafdan 2 ni ayirish.

$$6x^{2}-1+7x-2=0$$

-3 olish uchun -1 dan 2 ni ayirish.

$$6x^{2}-3+7x=0$$

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.

$$6x^{2}+7x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, 7 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\left(-3\right)}}{2\times 6}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 6\left(-3\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-24\left(-3\right)}}{2\times 6}$$

-24 ni -3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+72}}{2\times 6}$$

49 ni 72 ga qo'shish.

$$x=\frac{-7±\sqrt{121}}{2\times 6}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±11}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{-7±11}{12}$$

x=\frac{-7±11}{12} tenglamasini yeching, bunda ± musbat. -7 ni 11 ga qo'shish.

$$x=\frac{4}{12}$$

\frac{4}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{3}$$

x=\frac{-7±11}{12} tenglamasini yeching, bunda ± manfiy. -7 dan 11 ni ayirish.

$$x=-\frac{18}{12}$$

\frac{-18}{12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{3}{2}$$

Tenglama yechildi.

$$x=\frac{1}{3} x=-\frac{3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,-1,1,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right) ga, x^{2}-1,x^{4}-5x^{2}+4,4-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x^{2}-4\right)\times 4+15+7x=\left(-x^{2}+1\right)\times 2$$

x^{2}-4 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-16+15+7x=\left(-x^{2}+1\right)\times 2$$

-1 olish uchun -16 va 15'ni qo'shing.

$$4x^{2}-1+7x=\left(-x^{2}+1\right)\times 2$$

-x^{2}+1 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-1+7x=-2x^{2}+2$$

2x^{2} ni ikki tarafga qo’shing.

$$4x^{2}-1+7x+2x^{2}=2$$

6x^{2} ni olish uchun 4x^{2} va 2x^{2} ni birlashtirish.

$$6x^{2}-1+7x=2$$

1 ni ikki tarafga qo’shing.

$$6x^{2}+7x=2+1$$

3 olish uchun 2 va 1'ni qo'shing.

$$6x^{2}+7x=3$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}+7x}{6}=\frac{3}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{6}x=\frac{3}{6}$$

\frac{3}{6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{7}{6}x=\frac{1}{2}$$

\frac{7}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{12} olish uchun. Keyin, \frac{7}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=\frac{1}{2}+\left(\frac{7}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{12} kvadratini chiqarish.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{1}{2}+\frac{49}{144}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{49}{144} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{121}{144}$$

x^{2}+\frac{7}{6}x+\frac{49}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{121}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Qisqartirish.

$$x+\frac{7}{12}=\frac{11}{12} x+\frac{7}{12}=-\frac{11}{12}$$

Tenglamaning ikkala tarafidan \frac{7}{12} ni ayirish.

$$x=\frac{1}{3} x=-\frac{3}{2}$$