x-1/x+1=2x-4/x+4 eching

$$\frac{ x-1 }{ x+1 } = \frac{ 2x-4 }{ x+4 }$$

x uchun yechish

$$x=5$$

$$x=0$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -4,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(x+4\right) ga, x+1,x+4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

x+1 ga 2x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+3x-4=2x^{2}-2x-4$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}+3x-4-2x^{2}=-2x-4$$

-x^{2} ni olish uchun x^{2} va -2x^{2} ni birlashtirish.

$$-x^{2}+3x-4=-2x-4$$

2x ni ikki tarafga qo’shing.

$$-x^{2}+3x-4+2x=-4$$

5x ni olish uchun 3x va 2x ni birlashtirish.

$$-x^{2}+5x-4=-4$$

4 ni ikki tarafga qo’shing.

$$-x^{2}+5x-4+4=0$$

0 olish uchun -4 va 4'ni qo'shing.

$$-x^{2}+5x=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 5 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}}}{2\left(-1\right)}$$

5^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-5±5}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±5}{-2}$$

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

$$x=\frac{0}{-2}$$

0 ni -2 ga bo'lish.

$$x=0$$

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

$$x=-\frac{10}{-2}$$

-10 ni -2 ga bo'lish.

$$x=5$$

Tenglama yechildi.

$$x=0 x=5$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -4,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+1\right)\left(x+4\right) ga, x+1,x+4 ning eng kichik karralisiga ko‘paytiring.

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

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

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

x+1 ga 2x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+3x-4=2x^{2}-2x-4$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}+3x-4-2x^{2}=-2x-4$$

-x^{2} ni olish uchun x^{2} va -2x^{2} ni birlashtirish.

$$-x^{2}+3x-4=-2x-4$$

2x ni ikki tarafga qo’shing.

$$-x^{2}+3x-4+2x=-4$$

5x ni olish uchun 3x va 2x ni birlashtirish.

$$-x^{2}+5x-4=-4$$

4 ni ikki tarafga qo’shing.

$$-x^{2}+5x=-4+4$$

0 olish uchun -4 va 4'ni qo'shing.

$$-x^{2}+5x=0$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+5x}{-1}=\frac{0}{-1}$$

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

$$x^{2}+\frac{5}{-1}x=\frac{0}{-1}$$

5 ni -1 ga bo'lish.

$$x^{2}-5x=\frac{0}{-1}$$

0 ni -1 ga bo'lish.

$$x^{2}-5x=0$$

-5 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{2} olish uchun. Keyin, -\frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=\frac{25}{4}$$

x^{2}-5x+\frac{25}{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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{5}{2} x-\frac{5}{2}=-\frac{5}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=5 x=0$$