2x^3-12x^2+9x/4x^2(x^2+3)=(x-3)/2{x^2+6x} eching

$$\frac{ 2 { x }^{ 3 } -12 { x }^{ 2 } +9x }{ 4 { x }^{ 2 } ( { x }^{ 2 } +3) } = \frac{ (x-3) }{ 2 { x }^{ 2 } +6x }$$

x uchun yechish

$$x = \frac{15}{11} = 1\frac{4}{11} \approx 1,363636364$$

Faktorlash qadamlari

x qiymati -3,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+3\right)x^{2}\left(x^{2}+3\right) ga, 4x^{2}\left(x^{2}+3\right),2x^{2}+6x ning eng kichik karralisiga ko‘paytiring.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

x+3 ga 2x^{3}-12x^{2}+9x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

2x ga x^{2}+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

2x^{3}+6x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Ikkala tarafdan 2x^{4} ni ayirish.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

0 ni olish uchun 2x^{4} va -2x^{4} ni birlashtirish.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

6x^{3} ni ikki tarafga qo’shing.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

0 ni olish uchun -6x^{3} va 6x^{3} ni birlashtirish.

$$-27x^{2}+27x=6x^{2}-18x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-27x^{2}+27x-6x^{2}=-18x$$

-33x^{2} ni olish uchun -27x^{2} va -6x^{2} ni birlashtirish.

$$-33x^{2}+27x=-18x$$

18x ni ikki tarafga qo’shing.

$$-33x^{2}+27x+18x=0$$

45x ni olish uchun 27x va 18x ni birlashtirish.

$$-33x^{2}+45x=0$$

x omili.

$$x\left(-33x+45\right)=0$$

Tenglamani yechish uchun x=0 va -33x+45=0 ni yeching.

$$x=0 x=\frac{15}{11}$$

x qiymati 0 teng bo‘lmaydi.

$$x=\frac{15}{11}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+3\right)x^{2}\left(x^{2}+3\right) ga, 4x^{2}\left(x^{2}+3\right),2x^{2}+6x ning eng kichik karralisiga ko‘paytiring.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

x+3 ga 2x^{3}-12x^{2}+9x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

2x ga x^{2}+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

2x^{3}+6x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Ikkala tarafdan 2x^{4} ni ayirish.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

0 ni olish uchun 2x^{4} va -2x^{4} ni birlashtirish.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

6x^{3} ni ikki tarafga qo’shing.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

0 ni olish uchun -6x^{3} va 6x^{3} ni birlashtirish.

$$-27x^{2}+27x=6x^{2}-18x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-27x^{2}+27x-6x^{2}=-18x$$

-33x^{2} ni olish uchun -27x^{2} va -6x^{2} ni birlashtirish.

$$-33x^{2}+27x=-18x$$

18x ni ikki tarafga qo’shing.

$$-33x^{2}+27x+18x=0$$

45x ni olish uchun 27x va 18x ni birlashtirish.

$$-33x^{2}+45x=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -33 ni a, 45 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-45±\sqrt{45^{2}}}{2\left(-33\right)}$$

45^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-45±45}{2\left(-33\right)}$$

2 ni -33 marotabaga ko'paytirish.

$$x=\frac{-45±45}{-66}$$

x=\frac{-45±45}{-66} tenglamasini yeching, bunda ± musbat. -45 ni 45 ga qo'shish.

$$x=\frac{0}{-66}$$

0 ni -66 ga bo'lish.

$$x=0$$

x=\frac{-45±45}{-66} tenglamasini yeching, bunda ± manfiy. -45 dan 45 ni ayirish.

$$x=-\frac{90}{-66}$$

\frac{-90}{-66} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{15}{11}$$

Tenglama yechildi.

$$x=0 x=\frac{15}{11}$$

x qiymati 0 teng bo‘lmaydi.

$$x=\frac{15}{11}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x+3\right)x^{2}\left(x^{2}+3\right) ga, 4x^{2}\left(x^{2}+3\right),2x^{2}+6x ning eng kichik karralisiga ko‘paytiring.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

x+3 ga 2x^{3}-12x^{2}+9x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

2x ga x^{2}+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

2x^{3}+6x ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Ikkala tarafdan 2x^{4} ni ayirish.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

0 ni olish uchun 2x^{4} va -2x^{4} ni birlashtirish.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

6x^{3} ni ikki tarafga qo’shing.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

0 ni olish uchun -6x^{3} va 6x^{3} ni birlashtirish.

$$-27x^{2}+27x=6x^{2}-18x$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$-27x^{2}+27x-6x^{2}=-18x$$

-33x^{2} ni olish uchun -27x^{2} va -6x^{2} ni birlashtirish.

$$-33x^{2}+27x=-18x$$

18x ni ikki tarafga qo’shing.

$$-33x^{2}+27x+18x=0$$

45x ni olish uchun 27x va 18x ni birlashtirish.

$$-33x^{2}+45x=0$$

Ikki tarafini -33 ga bo‘ling.

$$\frac{-33x^{2}+45x}{-33}=\frac{0}{-33}$$

-33 ga bo'lish -33 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{45}{-33}x=\frac{0}{-33}$$

\frac{45}{-33} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{15}{11}x=\frac{0}{-33}$$

0 ni -33 ga bo'lish.

$$x^{2}-\frac{15}{11}x=0$$

-\frac{15}{11} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{22} olish uchun. Keyin, -\frac{15}{22} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{15}{11}x+\left(-\frac{15}{22}\right)^{2}=\left(-\frac{15}{22}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{22} kvadratini chiqarish.

$$x^{2}-\frac{15}{11}x+\frac{225}{484}=\frac{225}{484}$$

x^{2}-\frac{15}{11}x+\frac{225}{484} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{22}\right)^{2}=\frac{225}{484}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{22}\right)^{2}}=\sqrt{\frac{225}{484}}$$

Qisqartirish.

$$x-\frac{15}{22}=\frac{15}{22} x-\frac{15}{22}=-\frac{15}{22}$$

\frac{15}{22} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15}{11} x=0$$

x qiymati 0 teng bo‘lmaydi.

$$x=\frac{15}{11}$$