u+2/u-4-1=u+1/u-3 eching

$$\frac { u + 2 } { u - 4 } - 1 = \frac { u + 1 } { u - 3 }$$

u uchun yechish

$$u=2$$

$$u=7$$

Kvadrat tenglamalaridan foydalanish qoidalari

u qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(u-4\right)\left(u-3\right) ga, u-4,u-3 ning eng kichik karralisiga ko‘paytiring.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u-3 ga u+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u-4 ga u-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u^{2}-7u+12 ga -1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

0 ni olish uchun u^{2} va -u^{2} ni birlashtirish.

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

6u ni olish uchun -u va 7u ni birlashtirish.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

-18 olish uchun -6 dan 12 ni ayirish.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

u-4 ga u+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6u-18=u^{2}-3u-4$$

Ikkala tarafdan u^{2} ni ayirish.

$$6u-18-u^{2}=-3u-4$$

3u ni ikki tarafga qo’shing.

$$6u-18-u^{2}+3u=-4$$

9u ni olish uchun 6u va 3u ni birlashtirish.

$$9u-18-u^{2}=-4$$

4 ni ikki tarafga qo’shing.

$$9u-18-u^{2}+4=0$$

-14 olish uchun -18 va 4'ni qo'shing.

$$9u-14-u^{2}=0$$

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.

$$-u^{2}+9u-14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 9 ni b va -14 ni c bilan almashtiring.

$$u=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

9 kvadratini chiqarish.

$$u=\frac{-9±\sqrt{81-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$u=\frac{-9±\sqrt{81+4\left(-14\right)}}{2\left(-1\right)}$$

4 ni -14 marotabaga ko'paytirish.

$$u=\frac{-9±\sqrt{81-56}}{2\left(-1\right)}$$

81 ni -56 ga qo'shish.

$$u=\frac{-9±\sqrt{25}}{2\left(-1\right)}$$

25 ning kvadrat ildizini chiqarish.

$$u=\frac{-9±5}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$u=\frac{-9±5}{-2}$$

u=\frac{-9±5}{-2} tenglamasini yeching, bunda ± musbat. -9 ni 5 ga qo'shish.

$$u=-\frac{4}{-2}$$

-4 ni -2 ga bo'lish.

$$u=2$$

u=\frac{-9±5}{-2} tenglamasini yeching, bunda ± manfiy. -9 dan 5 ni ayirish.

$$u=-\frac{14}{-2}$$

-14 ni -2 ga bo'lish.

$$u=7$$

Tenglama yechildi.

$$u=2 u=7$$

Kvadratni yakunlash bo'yicha qadamlar

u qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(u-4\right)\left(u-3\right) ga, u-4,u-3 ning eng kichik karralisiga ko‘paytiring.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u-3 ga u+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u-4 ga u-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

u^{2}-7u+12 ga -1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

0 ni olish uchun u^{2} va -u^{2} ni birlashtirish.

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

6u ni olish uchun -u va 7u ni birlashtirish.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

-18 olish uchun -6 dan 12 ni ayirish.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

u-4 ga u+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6u-18=u^{2}-3u-4$$

Ikkala tarafdan u^{2} ni ayirish.

$$6u-18-u^{2}=-3u-4$$

3u ni ikki tarafga qo’shing.

$$6u-18-u^{2}+3u=-4$$

9u ni olish uchun 6u va 3u ni birlashtirish.

$$9u-18-u^{2}=-4$$

18 ni ikki tarafga qo’shing.

$$9u-u^{2}=-4+18$$

14 olish uchun -4 va 18'ni qo'shing.

$$9u-u^{2}=14$$

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

$$-u^{2}+9u=14$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-u^{2}+9u}{-1}=\frac{14}{-1}$$

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

$$u^{2}+\frac{9}{-1}u=\frac{14}{-1}$$

9 ni -1 ga bo'lish.

$$u^{2}-9u=\frac{14}{-1}$$

14 ni -1 ga bo'lish.

$$u^{2}-9u=-14$$

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

$$u^{2}-9u+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$u^{2}-9u+\frac{81}{4}=-14+\frac{81}{4}$$

-14 ni \frac{81}{4} ga qo'shish.

$$u^{2}-9u+\frac{81}{4}=\frac{25}{4}$$

u^{2}-9u+\frac{81}{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.

$$\left(u-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(u-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$u-\frac{9}{2}=\frac{5}{2} u-\frac{9}{2}=-\frac{5}{2}$$

\frac{9}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$u=7 u=2$$