14*x/12+x*14/12+x=4 eching

$$14 \times \frac{ x }{ 12+x } \times \frac{ 14 }{ 12+x } =4$$

x uchun yechish

$$x=9$$

$$x=16$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -12 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+12 ga ko'paytirish.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

14\times \frac{14}{12+x} ni yagona kasrga aylantiring.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

4 ga x+12 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{14\times 14}{12+x}x=4x+48$$

196 hosil qilish uchun 14 va 14 ni ko'paytirish.

$$\frac{196}{12+x}x=4x+48$$

\frac{196}{12+x}x ni yagona kasrga aylantiring.

$$\frac{196x}{12+x}=4x+48$$

Ikkala tarafdan 4x ni ayirish.

$$\frac{196x}{12+x}-4x=48$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -4x ni \frac{12+x}{12+x} marotabaga ko'paytirish.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

\frac{196x}{12+x} va \frac{-4x\left(12+x\right)}{12+x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

196x-4x\left(12+x\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

196x-48x-4x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{148x-4x^{2}}{12+x}=48$$

Ikkala tarafdan 48 ni ayirish.

$$\frac{148x-4x^{2}}{12+x}-48=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 48 ni \frac{12+x}{12+x} marotabaga ko'paytirish.

$$\frac{148x-4x^{2}}{12+x}-\frac{48\left(12+x\right)}{12+x}=0$$

\frac{148x-4x^{2}}{12+x} va \frac{48\left(12+x\right)}{12+x} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{148x-4x^{2}-48\left(12+x\right)}{12+x}=0$$

148x-4x^{2}-48\left(12+x\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{148x-4x^{2}-576-48x}{12+x}=0$$

148x-4x^{2}-576-48x kabi iboralarga o‘xshab birlashtiring.

$$\frac{100x-4x^{2}-576}{12+x}=0$$

x qiymati -12 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+12 ga ko'paytirish.

$$100x-4x^{2}-576=0$$

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.

$$-4x^{2}+100x-576=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 100 ni b va -576 ni c bilan almashtiring.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

100 kvadratini chiqarish.

$$x=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000+16\left(-576\right)}}{2\left(-4\right)}$$

16 ni -576 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000-9216}}{2\left(-4\right)}$$

10000 ni -9216 ga qo'shish.

$$x=\frac{-100±\sqrt{784}}{2\left(-4\right)}$$

784 ning kvadrat ildizini chiqarish.

$$x=\frac{-100±28}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-100±28}{-8}$$

x=\frac{-100±28}{-8} tenglamasini yeching, bunda ± musbat. -100 ni 28 ga qo'shish.

$$x=-\frac{72}{-8}$$

-72 ni -8 ga bo'lish.

$$x=9$$

x=\frac{-100±28}{-8} tenglamasini yeching, bunda ± manfiy. -100 dan 28 ni ayirish.

$$x=-\frac{128}{-8}$$

-128 ni -8 ga bo'lish.

$$x=16$$

Tenglama yechildi.

$$x=9 x=16$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -12 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+12 ga ko'paytirish.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

14\times \frac{14}{12+x} ni yagona kasrga aylantiring.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

4 ga x+12 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{14\times 14}{12+x}x=4x+48$$

196 hosil qilish uchun 14 va 14 ni ko'paytirish.

$$\frac{196}{12+x}x=4x+48$$

\frac{196}{12+x}x ni yagona kasrga aylantiring.

$$\frac{196x}{12+x}=4x+48$$

Ikkala tarafdan 4x ni ayirish.

$$\frac{196x}{12+x}-4x=48$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -4x ni \frac{12+x}{12+x} marotabaga ko'paytirish.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

\frac{196x}{12+x} va \frac{-4x\left(12+x\right)}{12+x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

196x-4x\left(12+x\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

196x-48x-4x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{148x-4x^{2}}{12+x}=48$$

x qiymati -12 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+12 ga ko'paytirish.

$$148x-4x^{2}=48\left(x+12\right)$$

48 ga x+12 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$148x-4x^{2}=48x+576$$

Ikkala tarafdan 48x ni ayirish.

$$148x-4x^{2}-48x=576$$

100x ni olish uchun 148x va -48x ni birlashtirish.

$$100x-4x^{2}=576$$

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

$$-4x^{2}+100x=576$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+100x}{-4}=\frac{576}{-4}$$

-4 ga bo'lish -4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{100}{-4}x=\frac{576}{-4}$$

100 ni -4 ga bo'lish.

$$x^{2}-25x=\frac{576}{-4}$$

576 ni -4 ga bo'lish.

$$x^{2}-25x=-144$$

-25 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{25}{2} olish uchun. Keyin, -\frac{25}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-144+\left(-\frac{25}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{25}{2} kvadratini chiqarish.

$$x^{2}-25x+\frac{625}{4}=-144+\frac{625}{4}$$

-144 ni \frac{625}{4} ga qo'shish.

$$x^{2}-25x+\frac{625}{4}=\frac{49}{4}$$

x^{2}-25x+\frac{625}{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.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{49}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Qisqartirish.

$$x-\frac{25}{2}=\frac{7}{2} x-\frac{25}{2}=-\frac{7}{2}$$

\frac{25}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=16 x=9$$