x+3/x-3+x-6/x+6=11 eching

$$\frac { x + 3 } { x - 3 } + \frac { x - 6 } { x + 6 } = 11$$

x uchun yechish

$$x = \frac{\sqrt{1057} - 11}{6} \approx 3,585256069$$

$$x=\frac{-\sqrt{1057}-11}{6}\approx -7,251922736$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -6,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x+6\right) ga, x-3,x+6 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

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

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

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

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

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

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

0 ni olish uchun 9x va -9x ni birlashtirish.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

36 olish uchun 18 va 18'ni qo'shing.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

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

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

11x-33 ga x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+36=11x^{2}+33x-198$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$2x^{2}+36-11x^{2}=33x-198$$

-9x^{2} ni olish uchun 2x^{2} va -11x^{2} ni birlashtirish.

$$-9x^{2}+36=33x-198$$

Ikkala tarafdan 33x ni ayirish.

$$-9x^{2}+36-33x=-198$$

198 ni ikki tarafga qo’shing.

$$-9x^{2}+36-33x+198=0$$

234 olish uchun 36 va 198'ni qo'shing.

$$-9x^{2}+234-33x=0$$

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.

$$-9x^{2}-33x+234=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, -33 ni b va 234 ni c bilan almashtiring.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

-33 kvadratini chiqarish.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-33\right)±\sqrt{1089+36\times 234}}{2\left(-9\right)}$$

36 ni 234 marotabaga ko'paytirish.

$$x=\frac{-\left(-33\right)±\sqrt{1089+8424}}{2\left(-9\right)}$$

1089 ni 8424 ga qo'shish.

$$x=\frac{-\left(-33\right)±\sqrt{9513}}{2\left(-9\right)}$$

9513 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-33\right)±3\sqrt{1057}}{2\left(-9\right)}$$

-33 ning teskarisi 33 ga teng.

$$x=\frac{33±3\sqrt{1057}}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{33±3\sqrt{1057}}{-18}$$

x=\frac{33±3\sqrt{1057}}{-18} tenglamasini yeching, bunda ± musbat. 33 ni 3\sqrt{1057} ga qo'shish.

$$x=\frac{3\sqrt{1057}+33}{-18}$$

33+3\sqrt{1057} ni -18 ga bo'lish.

$$x=\frac{-\sqrt{1057}-11}{6}$$

x=\frac{33±3\sqrt{1057}}{-18} tenglamasini yeching, bunda ± manfiy. 33 dan 3\sqrt{1057} ni ayirish.

$$x=\frac{33-3\sqrt{1057}}{-18}$$

33-3\sqrt{1057} ni -18 ga bo'lish.

$$x=\frac{\sqrt{1057}-11}{6}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{1057}-11}{6} x=\frac{\sqrt{1057}-11}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -6,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x+6\right) ga, x-3,x+6 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

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

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

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

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

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

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

0 ni olish uchun 9x va -9x ni birlashtirish.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

36 olish uchun 18 va 18'ni qo'shing.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

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

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

11x-33 ga x+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+36=11x^{2}+33x-198$$

Ikkala tarafdan 11x^{2} ni ayirish.

$$2x^{2}+36-11x^{2}=33x-198$$

-9x^{2} ni olish uchun 2x^{2} va -11x^{2} ni birlashtirish.

$$-9x^{2}+36=33x-198$$

Ikkala tarafdan 33x ni ayirish.

$$-9x^{2}+36-33x=-198$$

Ikkala tarafdan 36 ni ayirish.

$$-9x^{2}-33x=-198-36$$

-234 olish uchun -198 dan 36 ni ayirish.

$$-9x^{2}-33x=-234$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}-33x}{-9}=-\frac{234}{-9}$$

-9 ga bo'lish -9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{33}{-9}\right)x=-\frac{234}{-9}$$

\frac{-33}{-9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{11}{3}x=-\frac{234}{-9}$$

-234 ni -9 ga bo'lish.

$$x^{2}+\frac{11}{3}x=26$$

\frac{11}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{6} olish uchun. Keyin, \frac{11}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{11}{3}x+\left(\frac{11}{6}\right)^{2}=26+\left(\frac{11}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{6} kvadratini chiqarish.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=26+\frac{121}{36}$$

26 ni \frac{121}{36} ga qo'shish.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=\frac{1057}{36}$$

x^{2}+\frac{11}{3}x+\frac{121}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{11}{6}\right)^{2}=\frac{1057}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{6}\right)^{2}}=\sqrt{\frac{1057}{36}}$$

Qisqartirish.

$$x+\frac{11}{6}=\frac{\sqrt{1057}}{6} x+\frac{11}{6}=-\frac{\sqrt{1057}}{6}$$

Tenglamaning ikkala tarafidan \frac{11}{6} ni ayirish.

$$x=\frac{\sqrt{1057}-11}{6} x=\frac{-\sqrt{1057}-11}{6}$$