208/x+16+2=216/x eching

$$\frac { 208 } { x + 16 } + 2 = \frac { 216 } { x }$$

x uchun yechish

$$x=-48$$

$$x=36$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -16,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+16\right) ga, x+16,x ning eng kichik karralisiga ko‘paytiring.

$$x\times 208+x\left(x+16\right)\times 2=\left(x+16\right)\times 216$$

x ga x+16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 208+\left(x^{2}+16x\right)\times 2=\left(x+16\right)\times 216$$

x^{2}+16x ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 208+2x^{2}+32x=\left(x+16\right)\times 216$$

240x ni olish uchun x\times 208 va 32x ni birlashtirish.

$$240x+2x^{2}=\left(x+16\right)\times 216$$

x+16 ga 216 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$240x+2x^{2}=216x+3456$$

Ikkala tarafdan 216x ni ayirish.

$$240x+2x^{2}-216x=3456$$

24x ni olish uchun 240x va -216x ni birlashtirish.

$$24x+2x^{2}=3456$$

Ikkala tarafdan 3456 ni ayirish.

$$24x+2x^{2}-3456=0$$

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.

$$2x^{2}+24x-3456=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 24 ni b va -3456 ni c bilan almashtiring.

$$x=\frac{-24±\sqrt{24^{2}-4\times 2\left(-3456\right)}}{2\times 2}$$

24 kvadratini chiqarish.

$$x=\frac{-24±\sqrt{576-4\times 2\left(-3456\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-24±\sqrt{576-8\left(-3456\right)}}{2\times 2}$$

-8 ni -3456 marotabaga ko'paytirish.

$$x=\frac{-24±\sqrt{576+27648}}{2\times 2}$$

576 ni 27648 ga qo'shish.

$$x=\frac{-24±\sqrt{28224}}{2\times 2}$$

28224 ning kvadrat ildizini chiqarish.

$$x=\frac{-24±168}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-24±168}{4}$$

x=\frac{-24±168}{4} tenglamasini yeching, bunda ± musbat. -24 ni 168 ga qo'shish.

$$x=\frac{144}{4}$$

144 ni 4 ga bo'lish.

$$x=36$$

x=\frac{-24±168}{4} tenglamasini yeching, bunda ± manfiy. -24 dan 168 ni ayirish.

$$x=-\frac{192}{4}$$

-192 ni 4 ga bo'lish.

$$x=-48$$

Tenglama yechildi.

$$x=36 x=-48$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -16,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+16\right) ga, x+16,x ning eng kichik karralisiga ko‘paytiring.

$$x\times 208+x\left(x+16\right)\times 2=\left(x+16\right)\times 216$$

x ga x+16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 208+\left(x^{2}+16x\right)\times 2=\left(x+16\right)\times 216$$

x^{2}+16x ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 208+2x^{2}+32x=\left(x+16\right)\times 216$$

240x ni olish uchun x\times 208 va 32x ni birlashtirish.

$$240x+2x^{2}=\left(x+16\right)\times 216$$

x+16 ga 216 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$240x+2x^{2}=216x+3456$$

Ikkala tarafdan 216x ni ayirish.

$$240x+2x^{2}-216x=3456$$

24x ni olish uchun 240x va -216x ni birlashtirish.

$$24x+2x^{2}=3456$$

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

$$2x^{2}+24x=3456$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+24x}{2}=\frac{3456}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{24}{2}x=\frac{3456}{2}$$

24 ni 2 ga bo'lish.

$$x^{2}+12x=\frac{3456}{2}$$

3456 ni 2 ga bo'lish.

$$x^{2}+12x=1728$$

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

$$x^{2}+12x+6^{2}=1728+6^{2}$$

6 kvadratini chiqarish.

$$x^{2}+12x+36=1728+36$$

1728 ni 36 ga qo'shish.

$$x^{2}+12x+36=1764$$

x^{2}+12x+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.

$$\left(x+6\right)^{2}=1764$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{1764}$$

Qisqartirish.

$$x+6=42 x+6=-42$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=36 x=-48$$