10/(x+3)(x-7)-8/(x+3)(x-5)=x+10/(x-5)(x-7) eching

$$\frac { 10 } { ( x + 3 ) ( x - 7 ) } - \frac { 8 } { ( x + 3 ) ( x - 5 ) } = \frac { x + 10 } { ( x - 5 ) ( x - 7 ) }$$

x uchun yechish

$$x=-8$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,5,7 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-7\right)\left(x-5\right)\left(x+3\right) ga, \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right) ning eng kichik karralisiga ko‘paytiring.

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

x-5 ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

x-7 ga 8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

8x-56 teskarisini topish uchun har birining teskarisini toping.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

2x ni olish uchun 10x va -8x ni birlashtirish.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

6 olish uchun -50 va 56'ni qo'shing.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

x+3 ga x+10 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x+6=x^{2}+13x+30$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x+6-x^{2}=13x+30$$

Ikkala tarafdan 13x ni ayirish.

$$2x+6-x^{2}-13x=30$$

-11x ni olish uchun 2x va -13x ni birlashtirish.

$$-11x+6-x^{2}=30$$

Ikkala tarafdan 30 ni ayirish.

$$-11x+6-x^{2}-30=0$$

-24 olish uchun 6 dan 30 ni ayirish.

$$-11x-24-x^{2}=0$$

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.

$$-x^{2}-11x-24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -11 ni b va -24 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-24\right)}}{2\left(-1\right)}$$

4 ni -24 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-96}}{2\left(-1\right)}$$

121 ni -96 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{25}}{2\left(-1\right)}$$

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-11\right)±5}{2\left(-1\right)}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±5}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{11±5}{-2}$$

x=\frac{11±5}{-2} tenglamasini yeching, bunda ± musbat. 11 ni 5 ga qo'shish.

$$x=\frac{16}{-2}$$

16 ni -2 ga bo'lish.

$$x=-8$$

x=\frac{11±5}{-2} tenglamasini yeching, bunda ± manfiy. 11 dan 5 ni ayirish.

$$x=\frac{6}{-2}$$

6 ni -2 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=-8 x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=-8$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,5,7 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-7\right)\left(x-5\right)\left(x+3\right) ga, \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right) ning eng kichik karralisiga ko‘paytiring.

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

x-5 ga 10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

x-7 ga 8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

8x-56 teskarisini topish uchun har birining teskarisini toping.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

2x ni olish uchun 10x va -8x ni birlashtirish.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

6 olish uchun -50 va 56'ni qo'shing.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

x+3 ga x+10 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x+6=x^{2}+13x+30$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x+6-x^{2}=13x+30$$

Ikkala tarafdan 13x ni ayirish.

$$2x+6-x^{2}-13x=30$$

-11x ni olish uchun 2x va -13x ni birlashtirish.

$$-11x+6-x^{2}=30$$

Ikkala tarafdan 6 ni ayirish.

$$-11x-x^{2}=30-6$$

24 olish uchun 30 dan 6 ni ayirish.

$$-11x-x^{2}=24$$

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

$$-x^{2}-11x=24$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-11x}{-1}=\frac{24}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{24}{-1}$$

-11 ni -1 ga bo'lish.

$$x^{2}+11x=\frac{24}{-1}$$

24 ni -1 ga bo'lish.

$$x^{2}+11x=-24$$

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

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-24+\left(\frac{11}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{2} kvadratini chiqarish.

$$x^{2}+11x+\frac{121}{4}=-24+\frac{121}{4}$$

-24 ni \frac{121}{4} ga qo'shish.

$$x^{2}+11x+\frac{121}{4}=\frac{25}{4}$$

x^{2}+11x+\frac{121}{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.

$$\left(x+\frac{11}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x+\frac{11}{2}=\frac{5}{2} x+\frac{11}{2}=-\frac{5}{2}$$

Tenglamaning ikkala tarafidan \frac{11}{2} ni ayirish.

$$x=-3 x=-8$$

x qiymati -3 teng bo‘lmaydi.

$$x=-8$$