2200/100-x+15=22*100/67-x eching

$$\frac{ 2200 }{ 100-x } +15= \frac{ 22 \times 100 }{ 67-x }$$

x uchun yechish

$$x=12$$

$$x=155$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 67,100 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-100\right)\left(x-67\right) ga, 100-x,67-x ning eng kichik karralisiga ko‘paytiring.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

67-x ga 2200 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

x-100 ga x-67 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

x^{2}-167x+6700 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

-4705x ni olish uchun -2200x va -2505x ni birlashtirish.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

247900 olish uchun 147400 va 100500'ni qo'shing.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

2200 hosil qilish uchun 22 va 100 ni ko'paytirish.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

100-x ga 2200 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$247900-4705x+15x^{2}=220000-2200x$$

Ikkala tarafdan 220000 ni ayirish.

$$247900-4705x+15x^{2}-220000=-2200x$$

27900 olish uchun 247900 dan 220000 ni ayirish.

$$27900-4705x+15x^{2}=-2200x$$

2200x ni ikki tarafga qo’shing.

$$27900-4705x+15x^{2}+2200x=0$$

-2505x ni olish uchun -4705x va 2200x ni birlashtirish.

$$27900-2505x+15x^{2}=0$$

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.

$$15x^{2}-2505x+27900=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -2505 ni b va 27900 ni c bilan almashtiring.

$$x=\frac{-\left(-2505\right)±\sqrt{\left(-2505\right)^{2}-4\times 15\times 27900}}{2\times 15}$$

-2505 kvadratini chiqarish.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-4\times 15\times 27900}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-60\times 27900}}{2\times 15}$$

-60 ni 27900 marotabaga ko'paytirish.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-1674000}}{2\times 15}$$

6275025 ni -1674000 ga qo'shish.

$$x=\frac{-\left(-2505\right)±\sqrt{4601025}}{2\times 15}$$

4601025 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2505\right)±2145}{2\times 15}$$

-2505 ning teskarisi 2505 ga teng.

$$x=\frac{2505±2145}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{2505±2145}{30}$$

x=\frac{2505±2145}{30} tenglamasini yeching, bunda ± musbat. 2505 ni 2145 ga qo'shish.

$$x=\frac{4650}{30}$$

4650 ni 30 ga bo'lish.

$$x=155$$

x=\frac{2505±2145}{30} tenglamasini yeching, bunda ± manfiy. 2505 dan 2145 ni ayirish.

$$x=\frac{360}{30}$$

360 ni 30 ga bo'lish.

$$x=12$$

Tenglama yechildi.

$$x=155 x=12$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 67,100 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-100\right)\left(x-67\right) ga, 100-x,67-x ning eng kichik karralisiga ko‘paytiring.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

67-x ga 2200 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

x-100 ga x-67 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

x^{2}-167x+6700 ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

-4705x ni olish uchun -2200x va -2505x ni birlashtirish.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

247900 olish uchun 147400 va 100500'ni qo'shing.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

2200 hosil qilish uchun 22 va 100 ni ko'paytirish.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

100-x ga 2200 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$247900-4705x+15x^{2}=220000-2200x$$

2200x ni ikki tarafga qo’shing.

$$247900-4705x+15x^{2}+2200x=220000$$

-2505x ni olish uchun -4705x va 2200x ni birlashtirish.

$$247900-2505x+15x^{2}=220000$$

Ikkala tarafdan 247900 ni ayirish.

$$-2505x+15x^{2}=220000-247900$$

-27900 olish uchun 220000 dan 247900 ni ayirish.

$$-2505x+15x^{2}=-27900$$

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

$$15x^{2}-2505x=-27900$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-2505x}{15}=-\frac{27900}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{2505}{15}\right)x=-\frac{27900}{15}$$

-2505 ni 15 ga bo'lish.

$$x^{2}-167x=-\frac{27900}{15}$$

-27900 ni 15 ga bo'lish.

$$x^{2}-167x=-1860$$

-167 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{167}{2} olish uchun. Keyin, -\frac{167}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-167x+\left(-\frac{167}{2}\right)^{2}=-1860+\left(-\frac{167}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{167}{2} kvadratini chiqarish.

$$x^{2}-167x+\frac{27889}{4}=-1860+\frac{27889}{4}$$

-1860 ni \frac{27889}{4} ga qo'shish.

$$x^{2}-167x+\frac{27889}{4}=\frac{20449}{4}$$

x^{2}-167x+\frac{27889}{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.

$$\left(x-\frac{167}{2}\right)^{2}=\frac{20449}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{167}{2}\right)^{2}}=\sqrt{\frac{20449}{4}}$$

Qisqartirish.

$$x-\frac{167}{2}=\frac{143}{2} x-\frac{167}{2}=-\frac{143}{2}$$

\frac{167}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=155 x=12$$