1/2-x-1=1/x-2-5-x/3x^2-12 eching

$$\frac{ 1 }{ 2-x } -1 = \frac{ 1 }{ x-2 } - \frac{ 5-x }{ 3 { x }^{ 2 } -12 }$$

x uchun yechish

$$x=\frac{\sqrt{109}-7}{6}\approx 0,573384418$$

$$x=\frac{-\sqrt{109}-7}{6}\approx -2,906717751$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(x-2\right)\left(x+2\right) ga, 2-x,x-2,3x^{2}-12 ning eng kichik karralisiga ko‘paytiring.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(5-x\right)$$

-3 hosil qilish uchun 3 va -1 ni ko'paytirish.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

-3 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

-3x+6 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-6-3x-3x^{2}+12=3x+6-\left(5-x\right)$$

6 olish uchun -6 va 12'ni qo'shing.

$$6-3x-3x^{2}=3x+6-\left(5-x\right)$$

5-x teskarisini topish uchun har birining teskarisini toping.

$$6-3x-3x^{2}=3x+6-5+x$$

1 olish uchun 6 dan 5 ni ayirish.

$$6-3x-3x^{2}=3x+1+x$$

4x ni olish uchun 3x va x ni birlashtirish.

$$6-3x-3x^{2}=4x+1$$

Ikkala tarafdan 4x ni ayirish.

$$6-3x-3x^{2}-4x=1$$

-7x ni olish uchun -3x va -4x ni birlashtirish.

$$6-7x-3x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$6-7x-3x^{2}-1=0$$

5 olish uchun 6 dan 1 ni ayirish.

$$5-7x-3x^{2}=0$$

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.

$$-3x^{2}-7x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -7 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 5}}{2\left(-3\right)}$$

12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+60}}{2\left(-3\right)}$$

49 ni 60 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{109}}{2\left(-3\right)}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{109}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{109}}{-6}$$

x=\frac{7±\sqrt{109}}{-6} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{109} ga qo'shish.

$$x=\frac{\sqrt{109}+7}{-6}$$

7+\sqrt{109} ni -6 ga bo'lish.

$$x=\frac{-\sqrt{109}-7}{6}$$

x=\frac{7±\sqrt{109}}{-6} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{109} ni ayirish.

$$x=\frac{7-\sqrt{109}}{-6}$$

7-\sqrt{109} ni -6 ga bo'lish.

$$x=\frac{\sqrt{109}-7}{6}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{109}-7}{6} x=\frac{\sqrt{109}-7}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -2,2 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 3\left(x-2\right)\left(x+2\right) ga, 2-x,x-2,3x^{2}-12 ning eng kichik karralisiga ko‘paytiring.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(5-x\right)$$

-3 hosil qilish uchun 3 va -1 ni ko'paytirish.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

-3 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

-3x+6 ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-6-3x-3x^{2}+12=3x+6-\left(5-x\right)$$

6 olish uchun -6 va 12'ni qo'shing.

$$6-3x-3x^{2}=3x+6-\left(5-x\right)$$

5-x teskarisini topish uchun har birining teskarisini toping.

$$6-3x-3x^{2}=3x+6-5+x$$

1 olish uchun 6 dan 5 ni ayirish.

$$6-3x-3x^{2}=3x+1+x$$

4x ni olish uchun 3x va x ni birlashtirish.

$$6-3x-3x^{2}=4x+1$$

Ikkala tarafdan 4x ni ayirish.

$$6-3x-3x^{2}-4x=1$$

-7x ni olish uchun -3x va -4x ni birlashtirish.

$$6-7x-3x^{2}=1$$

Ikkala tarafdan 6 ni ayirish.

$$-7x-3x^{2}=1-6$$

-5 olish uchun 1 dan 6 ni ayirish.

$$-7x-3x^{2}=-5$$

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

$$-3x^{2}-7x=-5$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-7x}{-3}=-\frac{5}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{5}{-3}$$

-7 ni -3 ga bo'lish.

$$x^{2}+\frac{7}{3}x=-\frac{5}{-3}$$

-5 ni -3 ga bo'lish.

$$x^{2}+\frac{7}{3}x=\frac{5}{3}$$

\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{6} olish uchun. Keyin, \frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=\frac{5}{3}+\left(\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{6} kvadratini chiqarish.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{5}{3}+\frac{49}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{3} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{109}{36}$$

x^{2}+\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{109}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{109}{36}}$$

Qisqartirish.

$$x+\frac{7}{6}=\frac{\sqrt{109}}{6} x+\frac{7}{6}=-\frac{\sqrt{109}}{6}$$

Tenglamaning ikkala tarafidan \frac{7}{6} ni ayirish.

$$x=\frac{\sqrt{109}-7}{6} x=\frac{-\sqrt{109}-7}{6}$$