3x-7/x+5=x-3/x+2 eching

$$\frac { 3 x - 7 } { x + 5 } = \frac { x - 3 } { x + 2 }$$

x uchun yechish

$$x=\frac{1}{2}=0,5$$

$$x=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x+2\right)\left(3x-7\right)=\left(x+5\right)\left(x-3\right)$$

x+2 ga 3x-7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-x-14=\left(x+5\right)\left(x-3\right)$$

x+5 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-x-14=x^{2}+2x-15$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x^{2}-x-14-x^{2}=2x-15$$

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

$$2x^{2}-x-14=2x-15$$

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}-x-14-2x=-15$$

-3x ni olish uchun -x va -2x ni birlashtirish.

$$2x^{2}-3x-14=-15$$

15 ni ikki tarafga qo’shing.

$$2x^{2}-3x-14+15=0$$

1 olish uchun -14 va 15'ni qo'shing.

$$2x^{2}-3x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va 1 ni c bilan almashtiring.

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

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8}}{2\times 2}$$

9 ni -8 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{1}}{2\times 2}$$

1 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±1}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±1}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±1}{4}$$

x=\frac{3±1}{4} tenglamasini yeching, bunda ± musbat. 3 ni 1 ga qo'shish.

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

4 ni 4 ga bo'lish.

$$x=1$$

x=\frac{3±1}{4} tenglamasini yeching, bunda ± manfiy. 3 dan 1 ni ayirish.

$$x=\frac{2}{4}$$

\frac{2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x+2\right)\left(3x-7\right)=\left(x+5\right)\left(x-3\right)$$

x+2 ga 3x-7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-x-14=\left(x+5\right)\left(x-3\right)$$

x+5 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-x-14=x^{2}+2x-15$$

Ikkala tarafdan x^{2} ni ayirish.

$$3x^{2}-x-14-x^{2}=2x-15$$

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

$$2x^{2}-x-14=2x-15$$

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}-x-14-2x=-15$$

-3x ni olish uchun -x va -2x ni birlashtirish.

$$2x^{2}-3x-14=-15$$

14 ni ikki tarafga qo’shing.

$$2x^{2}-3x=-15+14$$

-1 olish uchun -15 va 14'ni qo'shing.

$$2x^{2}-3x=-1$$

Ikki tarafini 2 ga bo‘ling.

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

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

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

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{1}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Qisqartirish.

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

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

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