2400/x-50/x+15=9 eching

$$\frac { 2400 } { x } - \frac { 50 } { x + 15 } = 9$$

x uchun yechish

$$x = \frac{5 \sqrt{248089} + 2215}{18} \approx 261,412592793$$

$$x=\frac{2215-5\sqrt{248089}}{18}\approx -15,301481682$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -15,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+15\right) ga, x,x+15 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

x+15 ga 2400 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

9x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Ikkala tarafdan 135x ni ayirish.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

2265x ni olish uchun 2400x va -135x ni birlashtirish.

$$2265x+36000-x\times 50-9x^{2}=0$$

-50 hosil qilish uchun -1 va 50 ni ko'paytirish.

$$2265x+36000-50x-9x^{2}=0$$

2215x ni olish uchun 2265x va -50x ni birlashtirish.

$$2215x+36000-9x^{2}=0$$

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.

$$-9x^{2}+2215x+36000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, 2215 ni b va 36000 ni c bilan almashtiring.

$$x=\frac{-2215±\sqrt{2215^{2}-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

2215 kvadratini chiqarish.

$$x=\frac{-2215±\sqrt{4906225-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-2215±\sqrt{4906225+36\times 36000}}{2\left(-9\right)}$$

36 ni 36000 marotabaga ko'paytirish.

$$x=\frac{-2215±\sqrt{4906225+1296000}}{2\left(-9\right)}$$

4906225 ni 1296000 ga qo'shish.

$$x=\frac{-2215±\sqrt{6202225}}{2\left(-9\right)}$$

6202225 ning kvadrat ildizini chiqarish.

$$x=\frac{-2215±5\sqrt{248089}}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{-2215±5\sqrt{248089}}{-18}$$

x=\frac{-2215±5\sqrt{248089}}{-18} tenglamasini yeching, bunda ± musbat. -2215 ni 5\sqrt{248089} ga qo'shish.

$$x=\frac{5\sqrt{248089}-2215}{-18}$$

-2215+5\sqrt{248089} ni -18 ga bo'lish.

$$x=\frac{2215-5\sqrt{248089}}{18}$$

x=\frac{-2215±5\sqrt{248089}}{-18} tenglamasini yeching, bunda ± manfiy. -2215 dan 5\sqrt{248089} ni ayirish.

$$x=\frac{-5\sqrt{248089}-2215}{-18}$$

-2215-5\sqrt{248089} ni -18 ga bo'lish.

$$x=\frac{5\sqrt{248089}+2215}{18}$$

Tenglama yechildi.

$$x=\frac{2215-5\sqrt{248089}}{18} x=\frac{5\sqrt{248089}+2215}{18}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -15,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+15\right) ga, x,x+15 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

x+15 ga 2400 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

9x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Ikkala tarafdan 135x ni ayirish.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

2265x ni olish uchun 2400x va -135x ni birlashtirish.

$$2265x+36000-x\times 50-9x^{2}=0$$

Ikkala tarafdan 36000 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$2265x-x\times 50-9x^{2}=-36000$$

-50 hosil qilish uchun -1 va 50 ni ko'paytirish.

$$2265x-50x-9x^{2}=-36000$$

2215x ni olish uchun 2265x va -50x ni birlashtirish.

$$2215x-9x^{2}=-36000$$

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

$$-9x^{2}+2215x=-36000$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}+2215x}{-9}=-\frac{36000}{-9}$$

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

$$x^{2}+\frac{2215}{-9}x=-\frac{36000}{-9}$$

2215 ni -9 ga bo'lish.

$$x^{2}-\frac{2215}{9}x=-\frac{36000}{-9}$$

-36000 ni -9 ga bo'lish.

$$x^{2}-\frac{2215}{9}x=4000$$

-\frac{2215}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2215}{18} olish uchun. Keyin, -\frac{2215}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2215}{9}x+\left(-\frac{2215}{18}\right)^{2}=4000+\left(-\frac{2215}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2215}{18} kvadratini chiqarish.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=4000+\frac{4906225}{324}$$

4000 ni \frac{4906225}{324} ga qo'shish.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=\frac{6202225}{324}$$

x^{2}-\frac{2215}{9}x+\frac{4906225}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2215}{18}\right)^{2}=\frac{6202225}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2215}{18}\right)^{2}}=\sqrt{\frac{6202225}{324}}$$

Qisqartirish.

$$x-\frac{2215}{18}=\frac{5\sqrt{248089}}{18} x-\frac{2215}{18}=-\frac{5\sqrt{248089}}{18}$$

\frac{2215}{18} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5\sqrt{248089}+2215}{18} x=\frac{2215-5\sqrt{248089}}{18}$$