60/x+10+60/x-10=8 eching

$$\frac { 60 } { x + 10 } + \frac { 60 } { x - 10 } = 8$$

x uchun yechish

$$x=-5$$

$$x=20$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -10,10 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-10\right)\left(x+10\right) ga, x+10,x-10 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

x-10 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

x+10 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

120x ni olish uchun 60x va 60x ni birlashtirish.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

0 olish uchun -600 va 600'ni qo'shing.

$$120x=8\left(x-10\right)\left(x+10\right)$$

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

$$120x=\left(8x-80\right)\left(x+10\right)$$

8x-80 ga x+10 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$120x=8x^{2}-800$$

Ikkala tarafdan 8x^{2} ni ayirish.

$$120x-8x^{2}=-800$$

800 ni ikki tarafga qo’shing.

$$120x-8x^{2}+800=0$$

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.

$$-8x^{2}+120x+800=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 120 ni b va 800 ni c bilan almashtiring.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

120 kvadratini chiqarish.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400+32\times 800}}{2\left(-8\right)}$$

32 ni 800 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400+25600}}{2\left(-8\right)}$$

14400 ni 25600 ga qo'shish.

$$x=\frac{-120±\sqrt{40000}}{2\left(-8\right)}$$

40000 ning kvadrat ildizini chiqarish.

$$x=\frac{-120±200}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±200}{-16}$$

x=\frac{-120±200}{-16} tenglamasini yeching, bunda ± musbat. -120 ni 200 ga qo'shish.

$$x=\frac{80}{-16}$$

80 ni -16 ga bo'lish.

$$x=-5$$

x=\frac{-120±200}{-16} tenglamasini yeching, bunda ± manfiy. -120 dan 200 ni ayirish.

$$x=-\frac{320}{-16}$$

-320 ni -16 ga bo'lish.

$$x=20$$

Tenglama yechildi.

$$x=-5 x=20$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -10,10 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-10\right)\left(x+10\right) ga, x+10,x-10 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

x-10 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

x+10 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

120x ni olish uchun 60x va 60x ni birlashtirish.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

0 olish uchun -600 va 600'ni qo'shing.

$$120x=8\left(x-10\right)\left(x+10\right)$$

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

$$120x=\left(8x-80\right)\left(x+10\right)$$

8x-80 ga x+10 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$120x=8x^{2}-800$$

Ikkala tarafdan 8x^{2} ni ayirish.

$$120x-8x^{2}=-800$$

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

$$-8x^{2}+120x=-800$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+120x}{-8}=-\frac{800}{-8}$$

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

$$x^{2}+\frac{120}{-8}x=-\frac{800}{-8}$$

120 ni -8 ga bo'lish.

$$x^{2}-15x=-\frac{800}{-8}$$

-800 ni -8 ga bo'lish.

$$x^{2}-15x=100$$

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

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=100+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=100+\frac{225}{4}$$

100 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{625}{4}$$

x^{2}-15x+\frac{225}{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.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{625}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{25}{2} x-\frac{15}{2}=-\frac{25}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=20 x=-5$$