4/x+3+3/2x-1=1 eching

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

x uchun yechish

$$x=-1$$

$$x=4$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

2x-1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x-4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

x+3 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x-4+3x+9=\left(2x-1\right)\left(x+3\right)$$

11x ni olish uchun 8x va 3x ni birlashtirish.

$$11x-4+9=\left(2x-1\right)\left(x+3\right)$$

5 olish uchun -4 va 9'ni qo'shing.

$$11x+5=\left(2x-1\right)\left(x+3\right)$$

2x-1 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$11x+5=2x^{2}+5x-3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$11x+5-2x^{2}=5x-3$$

Ikkala tarafdan 5x ni ayirish.

$$11x+5-2x^{2}-5x=-3$$

6x ni olish uchun 11x va -5x ni birlashtirish.

$$6x+5-2x^{2}=-3$$

3 ni ikki tarafga qo’shing.

$$6x+5-2x^{2}+3=0$$

8 olish uchun 5 va 3'ni qo'shing.

$$6x+8-2x^{2}=0$$

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.

$$-2x^{2}+6x+8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 6 ni b va 8 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+8\times 8}}{2\left(-2\right)}$$

8 ni 8 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36+64}}{2\left(-2\right)}$$

36 ni 64 ga qo'shish.

$$x=\frac{-6±\sqrt{100}}{2\left(-2\right)}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±10}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-6±10}{-4}$$

x=\frac{-6±10}{-4} tenglamasini yeching, bunda ± musbat. -6 ni 10 ga qo'shish.

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

4 ni -4 ga bo'lish.

$$x=-1$$

x=\frac{-6±10}{-4} tenglamasini yeching, bunda ± manfiy. -6 dan 10 ni ayirish.

$$x=-\frac{16}{-4}$$

-16 ni -4 ga bo'lish.

$$x=4$$

Tenglama yechildi.

$$x=-1 x=4$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

2x-1 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x-4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

x+3 ga 3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x-4+3x+9=\left(2x-1\right)\left(x+3\right)$$

11x ni olish uchun 8x va 3x ni birlashtirish.

$$11x-4+9=\left(2x-1\right)\left(x+3\right)$$

5 olish uchun -4 va 9'ni qo'shing.

$$11x+5=\left(2x-1\right)\left(x+3\right)$$

2x-1 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$11x+5=2x^{2}+5x-3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$11x+5-2x^{2}=5x-3$$

Ikkala tarafdan 5x ni ayirish.

$$11x+5-2x^{2}-5x=-3$$

6x ni olish uchun 11x va -5x ni birlashtirish.

$$6x+5-2x^{2}=-3$$

Ikkala tarafdan 5 ni ayirish.

$$6x-2x^{2}=-3-5$$

-8 olish uchun -3 dan 5 ni ayirish.

$$6x-2x^{2}=-8$$

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

$$-2x^{2}+6x=-8$$

Ikki tarafini -2 ga bo‘ling.

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

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

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

6 ni -2 ga bo'lish.

$$x^{2}-3x=-\frac{8}{-2}$$

-8 ni -2 ga bo'lish.

$$x^{2}-3x=4$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}$$

4 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

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

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=-1$$