24/18-x-24/18+x=1 eching

$$\frac { 24 } { 18 - x } - \frac { 24 } { 18 + x } = 1$$

x uchun yechish

$$x=-54$$

$$x=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -18,18 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-18\right)\left(x+18\right) ga, 18-x,18+x ning eng kichik karralisiga ko‘paytiring.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

18+x teskarisini topish uchun har birining teskarisini toping.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

-18-x ga 24 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

x-18 ga 24 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

24x-432 teskarisini topish uchun har birining teskarisini toping.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

-48x ni olish uchun -24x va -24x ni birlashtirish.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

0 olish uchun -432 va 432'ni qo'shing.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Hisoblang: \left(x-18\right)\left(x+18\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 18 kvadratini chiqarish.

$$-48x=x^{2}-324$$

Ikkala tarafdan x^{2} ni ayirish.

$$-48x-x^{2}=-324$$

324 ni ikki tarafga qo’shing.

$$-48x-x^{2}+324=0$$

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.

$$-x^{2}-48x+324=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -48 ni b va 324 ni c bilan almashtiring.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

-48 kvadratini chiqarish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304+4\times 324}}{2\left(-1\right)}$$

4 ni 324 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304+1296}}{2\left(-1\right)}$$

2304 ni 1296 ga qo'shish.

$$x=\frac{-\left(-48\right)±\sqrt{3600}}{2\left(-1\right)}$$

3600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-48\right)±60}{2\left(-1\right)}$$

-48 ning teskarisi 48 ga teng.

$$x=\frac{48±60}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{48±60}{-2}$$

x=\frac{48±60}{-2} tenglamasini yeching, bunda ± musbat. 48 ni 60 ga qo'shish.

$$x=\frac{108}{-2}$$

108 ni -2 ga bo'lish.

$$x=-54$$

x=\frac{48±60}{-2} tenglamasini yeching, bunda ± manfiy. 48 dan 60 ni ayirish.

$$x=-\frac{12}{-2}$$

-12 ni -2 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=-54 x=6$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -18,18 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-18\right)\left(x+18\right) ga, 18-x,18+x ning eng kichik karralisiga ko‘paytiring.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

18+x teskarisini topish uchun har birining teskarisini toping.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

-18-x ga 24 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

x-18 ga 24 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

24x-432 teskarisini topish uchun har birining teskarisini toping.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

-48x ni olish uchun -24x va -24x ni birlashtirish.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

0 olish uchun -432 va 432'ni qo'shing.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Hisoblang: \left(x-18\right)\left(x+18\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 18 kvadratini chiqarish.

$$-48x=x^{2}-324$$

Ikkala tarafdan x^{2} ni ayirish.

$$-48x-x^{2}=-324$$

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

$$-x^{2}-48x=-324$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-48x}{-1}=-\frac{324}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{48}{-1}\right)x=-\frac{324}{-1}$$

-48 ni -1 ga bo'lish.

$$x^{2}+48x=-\frac{324}{-1}$$

-324 ni -1 ga bo'lish.

$$x^{2}+48x=324$$

48 ni bo‘lish, x shartining koeffitsienti, 2 ga 24 olish uchun. Keyin, 24 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+48x+24^{2}=324+24^{2}$$

24 kvadratini chiqarish.

$$x^{2}+48x+576=324+576$$

324 ni 576 ga qo'shish.

$$x^{2}+48x+576=900$$

x^{2}+48x+576 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+24\right)^{2}=900$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{900}$$

Qisqartirish.

$$x+24=30 x+24=-30$$

Tenglamaning ikkala tarafidan 24 ni ayirish.

$$x=6 x=-54$$