2x^2+1/x^2+3x-4=2(2)^2+1/2^2+3(2)-4= eching

$$\frac { 2 x ^ { 2 } + 1 } { x ^ { 2 } + 3 x - 4 } = \frac { 2 ( 2 ) ^ { 2 } + 1 } { 2 ^ { 2 } + 3 ( 2 ) - 4 } =$$

x uchun yechish

$$x=2$$

$$x=7$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -4,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+4\right) ga ko'paytirish.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(2\times 2^{2}+1\right)$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 2 ni qo‘shib, 3 ni oling.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(2^{3}+1\right)$$

3 daraja ko‘rsatkichini 2 ga hisoblang va 8 ni qiymatni oling.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(8+1\right)$$

9 olish uchun 8 va 1'ni qo'shing.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\times 9$$

\frac{3}{2} hosil qilish uchun \frac{1}{6} va 9 ni ko'paytirish.

$$2x^{2}+1=\frac{3}{2}\left(x-1\right)\left(x+4\right)$$

\frac{3}{2} ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+1=\left(\frac{3}{2}x-\frac{3}{2}\right)\left(x+4\right)$$

\frac{3}{2}x-\frac{3}{2} ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+1=\frac{3}{2}x^{2}+\frac{9}{2}x-6$$

Ikkala tarafdan \frac{3}{2}x^{2} ni ayirish.

$$2x^{2}+1-\frac{3}{2}x^{2}=\frac{9}{2}x-6$$

\frac{1}{2}x^{2} ni olish uchun 2x^{2} va -\frac{3}{2}x^{2} ni birlashtirish.

$$\frac{1}{2}x^{2}+1=\frac{9}{2}x-6$$

Ikkala tarafdan \frac{9}{2}x ni ayirish.

$$\frac{1}{2}x^{2}+1-\frac{9}{2}x=-6$$

6 ni ikki tarafga qo’shing.

$$\frac{1}{2}x^{2}+1-\frac{9}{2}x+6=0$$

7 olish uchun 1 va 6'ni qo'shing.

$$\frac{1}{2}x^{2}+7-\frac{9}{2}x=0$$

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.

$$\frac{1}{2}x^{2}-\frac{9}{2}x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{2} ni a, -\frac{9}{2} ni b va 7 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\left(-\frac{9}{2}\right)^{2}-4\times \frac{1}{2}\times 7}}{2\times \frac{1}{2}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-4\times \frac{1}{2}\times 7}}{2\times \frac{1}{2}}$$

-4 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-2\times 7}}{2\times \frac{1}{2}}$$

-2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-14}}{2\times \frac{1}{2}}$$

\frac{81}{4} ni -14 ga qo'shish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{25}{4}}}{2\times \frac{1}{2}}$$

\frac{25}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{9}{2}\right)±\frac{5}{2}}{2\times \frac{1}{2}}$$

-\frac{9}{2} ning teskarisi \frac{9}{2} ga teng.

$$x=\frac{\frac{9}{2}±\frac{5}{2}}{2\times \frac{1}{2}}$$

2 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{\frac{9}{2}±\frac{5}{2}}{1}$$

x=\frac{\frac{9}{2}±\frac{5}{2}}{1} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{2} ni \frac{5}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{7}{1}$$

7 ni 1 ga bo'lish.

$$x=7$$

x=\frac{\frac{9}{2}±\frac{5}{2}}{1} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{5}{2} ni \frac{9}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

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

2 ni 1 ga bo'lish.

$$x=2$$

Tenglama yechildi.

$$x=7 x=2$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -4,1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(x+4\right) ga ko'paytirish.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(2\times 2^{2}+1\right)$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 2 ni qo‘shib, 3 ni oling.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(2^{3}+1\right)$$

3 daraja ko‘rsatkichini 2 ga hisoblang va 8 ni qiymatni oling.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\left(8+1\right)$$

9 olish uchun 8 va 1'ni qo'shing.

$$2x^{2}+1=\frac{1}{6}\left(x-1\right)\left(x+4\right)\times 9$$

\frac{3}{2} hosil qilish uchun \frac{1}{6} va 9 ni ko'paytirish.

$$2x^{2}+1=\frac{3}{2}\left(x-1\right)\left(x+4\right)$$

\frac{3}{2} ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+1=\left(\frac{3}{2}x-\frac{3}{2}\right)\left(x+4\right)$$

\frac{3}{2}x-\frac{3}{2} ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+1=\frac{3}{2}x^{2}+\frac{9}{2}x-6$$

Ikkala tarafdan \frac{3}{2}x^{2} ni ayirish.

$$2x^{2}+1-\frac{3}{2}x^{2}=\frac{9}{2}x-6$$

\frac{1}{2}x^{2} ni olish uchun 2x^{2} va -\frac{3}{2}x^{2} ni birlashtirish.

$$\frac{1}{2}x^{2}+1=\frac{9}{2}x-6$$

Ikkala tarafdan \frac{9}{2}x ni ayirish.

$$\frac{1}{2}x^{2}+1-\frac{9}{2}x=-6$$

Ikkala tarafdan 1 ni ayirish.

$$\frac{1}{2}x^{2}-\frac{9}{2}x=-6-1$$

-7 olish uchun -6 dan 1 ni ayirish.

$$\frac{1}{2}x^{2}-\frac{9}{2}x=-7$$

Ikkala tarafini 2 ga ko‘paytiring.

$$\frac{\frac{1}{2}x^{2}-\frac{9}{2}x}{\frac{1}{2}}=-\frac{7}{\frac{1}{2}}$$

\frac{1}{2} ga bo'lish \frac{1}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{9}{2}}{\frac{1}{2}}\right)x=-\frac{7}{\frac{1}{2}}$$

-\frac{9}{2} ni \frac{1}{2} ga bo'lish -\frac{9}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-9x=-\frac{7}{\frac{1}{2}}$$

-7 ni \frac{1}{2} ga bo'lish -7 ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-9x=-14$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x^{2}-9x+\frac{81}{4}=-14+\frac{81}{4}$$

-14 ni \frac{81}{4} ga qo'shish.

$$x^{2}-9x+\frac{81}{4}=\frac{25}{4}$$

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

$$\left(x-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{9}{2}=\frac{5}{2} x-\frac{9}{2}=-\frac{5}{2}$$

\frac{9}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=7 x=2$$