4/x+3-5/3-x=1/x-3-1 eching

$$\frac { 4 } { x + 3 } - \frac { 5 } { 3 - x } = \frac { 1 } { x - 3 } - 1$$

x uchun yechish

$$x=-9$$

$$x=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\left(x-3\right)\times 4-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

x-3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x-12-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-5 hosil qilish uchun -1 va 5 ni ko'paytirish.

$$4x-12-\left(-5\left(3+x\right)\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-5 ga 3+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x-12-\left(-15-5x\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-15-5x teskarisini topish uchun har birining teskarisini toping.

$$4x-12+15+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

3 olish uchun -12 va 15'ni qo'shing.

$$4x+3+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

9x ni olish uchun 4x va 5x ni birlashtirish.

$$9x+3=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

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

$$9x+3=x+3+\left(x^{2}-9\right)\left(-1\right)$$

x^{2}-9 ga -1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$9x+3=x+3-x^{2}+9$$

12 olish uchun 3 va 9'ni qo'shing.

$$9x+3=x+12-x^{2}$$

Ikkala tarafdan x ni ayirish.

$$9x+3-x=12-x^{2}$$

8x ni olish uchun 9x va -x ni birlashtirish.

$$8x+3=12-x^{2}$$

Ikkala tarafdan 12 ni ayirish.

$$8x+3-12=-x^{2}$$

-9 olish uchun 3 dan 12 ni ayirish.

$$8x-9=-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$8x-9+x^{2}=0$$

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.

$$x^{2}+8x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 8 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-9\right)}}{2}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\left(-9\right)}}{2}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64+36}}{2}$$

64 ni 36 ga qo'shish.

$$x=\frac{-8±\sqrt{100}}{2}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±10}{2}$$

x=\frac{-8±10}{2} tenglamasini yeching, bunda ± musbat. -8 ni 10 ga qo'shish.

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

2 ni 2 ga bo'lish.

$$x=1$$

x=\frac{-8±10}{2} tenglamasini yeching, bunda ± manfiy. -8 dan 10 ni ayirish.

$$x=-\frac{18}{2}$$

-18 ni 2 ga bo'lish.

$$x=-9$$

Tenglama yechildi.

$$x=1 x=-9$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\left(x-3\right)\times 4-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

x-3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x-12-\left(-\left(3+x\right)\times 5\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-5 hosil qilish uchun -1 va 5 ni ko'paytirish.

$$4x-12-\left(-5\left(3+x\right)\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-5 ga 3+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x-12-\left(-15-5x\right)=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

-15-5x teskarisini topish uchun har birining teskarisini toping.

$$4x-12+15+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

3 olish uchun -12 va 15'ni qo'shing.

$$4x+3+5x=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

9x ni olish uchun 4x va 5x ni birlashtirish.

$$9x+3=x+3+\left(x-3\right)\left(x+3\right)\left(-1\right)$$

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

$$9x+3=x+3+\left(x^{2}-9\right)\left(-1\right)$$

x^{2}-9 ga -1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$9x+3=x+3-x^{2}+9$$

12 olish uchun 3 va 9'ni qo'shing.

$$9x+3=x+12-x^{2}$$

Ikkala tarafdan x ni ayirish.

$$9x+3-x=12-x^{2}$$

8x ni olish uchun 9x va -x ni birlashtirish.

$$8x+3=12-x^{2}$$

x^{2} ni ikki tarafga qo’shing.

$$8x+3+x^{2}=12$$

Ikkala tarafdan 3 ni ayirish.

$$8x+x^{2}=12-3$$

9 olish uchun 12 dan 3 ni ayirish.

$$8x+x^{2}=9$$

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

$$x^{2}+8x=9$$

8 ni bo‘lish, x shartining koeffitsienti, 2 ga 4 olish uchun. Keyin, 4 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+8x+4^{2}=9+4^{2}$$

4 kvadratini chiqarish.

$$x^{2}+8x+16=9+16$$

9 ni 16 ga qo'shish.

$$x^{2}+8x+16=25$$

x^{2}+8x+16 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+4\right)^{2}=25$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{25}$$

Qisqartirish.

$$x+4=5 x+4=-5$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$x=1 x=-9$$