1/(x-3)=2x+1/x^2-4 eching

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

x uchun yechish

$$x = \frac{\sqrt{21} + 5}{2} \approx 4,791287847$$

$$x=\frac{5-\sqrt{21}}{2}\approx 0,208712153$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x-3,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

x-3 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-4=2x^{2}-5x-3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-4-2x^{2}=-5x-3$$

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

$$-x^{2}-4=-5x-3$$

5x ni ikki tarafga qo’shing.

$$-x^{2}-4+5x=-3$$

3 ni ikki tarafga qo’shing.

$$-x^{2}-4+5x+3=0$$

-1 olish uchun -4 va 3'ni qo'shing.

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

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.

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

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 -1 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+4\left(-1\right)}}{2\left(-1\right)}$$

4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-4}}{2\left(-1\right)}$$

25 ni -4 ga qo'shish.

$$x=\frac{-5±\sqrt{21}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{21}}{-2}$$

x=\frac{-5±\sqrt{21}}{-2} tenglamasini yeching, bunda ± musbat. -5 ni \sqrt{21} ga qo'shish.

$$x=\frac{\sqrt{21}-5}{-2}$$

-5+\sqrt{21} ni -2 ga bo'lish.

$$x=\frac{5-\sqrt{21}}{2}$$

x=\frac{-5±\sqrt{21}}{-2} tenglamasini yeching, bunda ± manfiy. -5 dan \sqrt{21} ni ayirish.

$$x=\frac{-\sqrt{21}-5}{-2}$$

-5-\sqrt{21} ni -2 ga bo'lish.

$$x=\frac{\sqrt{21}+5}{2}$$

Tenglama yechildi.

$$x=\frac{5-\sqrt{21}}{2} x=\frac{\sqrt{21}+5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+2\right) ga, x-3,x^{2}-4 ning eng kichik karralisiga ko‘paytiring.

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

x-3 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-4=2x^{2}-5x-3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-4-2x^{2}=-5x-3$$

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

$$-x^{2}-4=-5x-3$$

5x ni ikki tarafga qo’shing.

$$-x^{2}-4+5x=-3$$

4 ni ikki tarafga qo’shing.

$$-x^{2}+5x=-3+4$$

1 olish uchun -3 va 4'ni qo'shing.

$$-x^{2}+5x=1$$

Ikki tarafini -1 ga bo‘ling.

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

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

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

5 ni -1 ga bo'lish.

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

1 ni -1 ga bo'lish.

$$x^{2}-5x=-1$$

-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.

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

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

$$x^{2}-5x+\frac{25}{4}=-1+\frac{25}{4}$$

-1 ni \frac{25}{4} ga qo'shish.

$$x^{2}-5x+\frac{25}{4}=\frac{21}{4}$$

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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{21}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{21}}{2} x-\frac{5}{2}=-\frac{\sqrt{21}}{2}$$

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

$$x=\frac{\sqrt{21}+5}{2} x=\frac{5-\sqrt{21}}{2}$$