(x+3)/(x-2)-8/3=(x+2)/(x-1) eching

$$\frac { ( x + 3 ) } { ( x - 2 ) } - \frac { 8 } { 3 } = \frac { ( x + 2 ) } { ( x - 1 ) }$$

x uchun yechish

$$x = \frac{13}{4} = 3\frac{1}{4} = 3,25$$

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

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(3x-3\right)\left(x+3\right)+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

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

$$3x^{2}+6x-9+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

-8 hosil qilish uchun 3 va -\frac{8}{3} ni ko'paytirish.

$$3x^{2}+6x-9-8\left(x-2\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

-8 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+6x-9+\left(-8x+16\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

-8x+16 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+6x-9-8x^{2}+24x-16=\left(3x-6\right)\left(x+2\right)$$

-5x^{2} ni olish uchun 3x^{2} va -8x^{2} ni birlashtirish.

$$-5x^{2}+6x-9+24x-16=\left(3x-6\right)\left(x+2\right)$$

30x ni olish uchun 6x va 24x ni birlashtirish.

$$-5x^{2}+30x-9-16=\left(3x-6\right)\left(x+2\right)$$

-25 olish uchun -9 dan 16 ni ayirish.

$$-5x^{2}+30x-25=\left(3x-6\right)\left(x+2\right)$$

3x-6 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-5x^{2}+30x-25=3x^{2}-12$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$-5x^{2}+30x-25-3x^{2}=-12$$

-8x^{2} ni olish uchun -5x^{2} va -3x^{2} ni birlashtirish.

$$-8x^{2}+30x-25=-12$$

12 ni ikki tarafga qo’shing.

$$-8x^{2}+30x-25+12=0$$

-13 olish uchun -25 va 12'ni qo'shing.

$$-8x^{2}+30x-13=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 30 ni b va -13 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-8\right)\left(-13\right)}}{2\left(-8\right)}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\left(-8\right)\left(-13\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+32\left(-13\right)}}{2\left(-8\right)}$$

32 ni -13 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-416}}{2\left(-8\right)}$$

900 ni -416 ga qo'shish.

$$x=\frac{-30±\sqrt{484}}{2\left(-8\right)}$$

484 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±22}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-30±22}{-16}$$

x=\frac{-30±22}{-16} tenglamasini yeching, bunda ± musbat. -30 ni 22 ga qo'shish.

$$x=-\frac{8}{-16}$$

\frac{-8}{-16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-30±22}{-16} tenglamasini yeching, bunda ± manfiy. -30 dan 22 ni ayirish.

$$x=-\frac{52}{-16}$$

\frac{-52}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{13}{4}$$

Tenglama yechildi.

$$x=\frac{1}{2} x=\frac{13}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(3x-3\right)\left(x+3\right)+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

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

$$3x^{2}+6x-9+3\left(x-2\right)\left(x-1\right)\left(-\frac{8}{3}\right)=\left(3x-6\right)\left(x+2\right)$$

-8 hosil qilish uchun 3 va -\frac{8}{3} ni ko'paytirish.

$$3x^{2}+6x-9-8\left(x-2\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

-8 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3x^{2}+6x-9+\left(-8x+16\right)\left(x-1\right)=\left(3x-6\right)\left(x+2\right)$$

-8x+16 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}+6x-9-8x^{2}+24x-16=\left(3x-6\right)\left(x+2\right)$$

-5x^{2} ni olish uchun 3x^{2} va -8x^{2} ni birlashtirish.

$$-5x^{2}+6x-9+24x-16=\left(3x-6\right)\left(x+2\right)$$

30x ni olish uchun 6x va 24x ni birlashtirish.

$$-5x^{2}+30x-9-16=\left(3x-6\right)\left(x+2\right)$$

-25 olish uchun -9 dan 16 ni ayirish.

$$-5x^{2}+30x-25=\left(3x-6\right)\left(x+2\right)$$

3x-6 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-5x^{2}+30x-25=3x^{2}-12$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$-5x^{2}+30x-25-3x^{2}=-12$$

-8x^{2} ni olish uchun -5x^{2} va -3x^{2} ni birlashtirish.

$$-8x^{2}+30x-25=-12$$

25 ni ikki tarafga qo’shing.

$$-8x^{2}+30x=-12+25$$

13 olish uchun -12 va 25'ni qo'shing.

$$-8x^{2}+30x=13$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+30x}{-8}=\frac{13}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{30}{-8}x=\frac{13}{-8}$$

\frac{30}{-8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{15}{4}x=\frac{13}{-8}$$

13 ni -8 ga bo'lish.

$$x^{2}-\frac{15}{4}x=-\frac{13}{8}$$

-\frac{15}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{8} olish uchun. Keyin, -\frac{15}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=-\frac{13}{8}+\left(-\frac{15}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{8} kvadratini chiqarish.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=-\frac{13}{8}+\frac{225}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{13}{8} ni \frac{225}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{121}{64}$$

x^{2}-\frac{15}{4}x+\frac{225}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{121}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Qisqartirish.

$$x-\frac{15}{8}=\frac{11}{8} x-\frac{15}{8}=-\frac{11}{8}$$

\frac{15}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{13}{4} x=\frac{1}{2}$$