4(x^2+1)(2x^2+1)=5(x^2-1)^2 eching

$$4 ( x ^ { 2 } + 1 ) ( 2 x ^ { 2 } + 1 ) = 5 ( x ^ { 2 } - 1 ) ^ { 2 }$$

x uchun yechish (complex solution)

$$x=\frac{i\sqrt{6\sqrt{31}+33}}{3}\approx 2,716341211i$$

$$x=-\frac{i\sqrt{6\sqrt{31}+33}}{3}\approx -0-2,716341211i$$

$$x=-\frac{\sqrt{6\sqrt{31}-33}}{3}\approx -0,212547035$$

$$x=\frac{\sqrt{6\sqrt{31}-33}}{3}\approx 0,212547035$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

$$8x^{4}+12x^{2}+4=5\left(x^{2}-1\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x^{2}-1\right)^{2} kengaytirilishi uchun ishlating.

$$8x^{4}+12x^{2}+4=5\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$8x^{4}+12x^{2}+4=5\left(x^{4}-2x^{2}+1\right)$$

5 ga x^{4}-2x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{4}+12x^{2}+4=5x^{4}-10x^{2}+5$$

Ikkala tarafdan 5x^{4} ni ayirish.

$$8x^{4}+12x^{2}+4-5x^{4}=-10x^{2}+5$$

3x^{4} ni olish uchun 8x^{4} va -5x^{4} ni birlashtirish.

$$3x^{4}+12x^{2}+4=-10x^{2}+5$$

10x^{2} ni ikki tarafga qo’shing.

$$3x^{4}+12x^{2}+4+10x^{2}=5$$

22x^{2} ni olish uchun 12x^{2} va 10x^{2} ni birlashtirish.

$$3x^{4}+22x^{2}+4=5$$

Ikkala tarafdan 5 ni ayirish.

$$3x^{4}+22x^{2}+4-5=0$$

-1 olish uchun 4 dan 5 ni ayirish.

$$3x^{4}+22x^{2}-1=0$$

x^{2} uchun t ni almashtiring.

$$3t^{2}+22t-1=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 3 ni, b uchun 22 ni va c uchun -1 ni ayiring.

$$t=\frac{-22±\sqrt{22^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Hisoblarni amalga oshiring.

$$t=\frac{-22±4\sqrt{31}}{6}$$

t=\frac{-22±4\sqrt{31}}{6} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=\frac{2\sqrt{31}-11}{3} t=\frac{-2\sqrt{31}-11}{3}$$

x=t^{2} boʻlganda, yechimlar har bir t uchun x=±\sqrt{t} hisoblanishi orqali olinadi.

$$x=-\sqrt{\frac{2\sqrt{31}-11}{3}} x=\sqrt{\frac{2\sqrt{31}-11}{3}} x=-i\sqrt{\frac{2\sqrt{31}+11}{3}} x=i\sqrt{\frac{2\sqrt{31}+11}{3}}$$

x uchun yechish

$$x=-\frac{\sqrt{6\sqrt{31}-33}}{3}\approx -0,212547035$$

$$x=\frac{\sqrt{6\sqrt{31}-33}}{3}\approx 0,212547035$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

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

$$8x^{4}+12x^{2}+4=5\left(x^{2}-1\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x^{2}-1\right)^{2} kengaytirilishi uchun ishlating.

$$8x^{4}+12x^{2}+4=5\left(\left(x^{2}\right)^{2}-2x^{2}+1\right)$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$8x^{4}+12x^{2}+4=5\left(x^{4}-2x^{2}+1\right)$$

5 ga x^{4}-2x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{4}+12x^{2}+4=5x^{4}-10x^{2}+5$$

Ikkala tarafdan 5x^{4} ni ayirish.

$$8x^{4}+12x^{2}+4-5x^{4}=-10x^{2}+5$$

3x^{4} ni olish uchun 8x^{4} va -5x^{4} ni birlashtirish.

$$3x^{4}+12x^{2}+4=-10x^{2}+5$$

10x^{2} ni ikki tarafga qo’shing.

$$3x^{4}+12x^{2}+4+10x^{2}=5$$

22x^{2} ni olish uchun 12x^{2} va 10x^{2} ni birlashtirish.

$$3x^{4}+22x^{2}+4=5$$

Ikkala tarafdan 5 ni ayirish.

$$3x^{4}+22x^{2}+4-5=0$$

-1 olish uchun 4 dan 5 ni ayirish.

$$3x^{4}+22x^{2}-1=0$$

x^{2} uchun t ni almashtiring.

$$3t^{2}+22t-1=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 3 ni, b uchun 22 ni va c uchun -1 ni ayiring.

$$t=\frac{-22±\sqrt{22^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Hisoblarni amalga oshiring.

$$t=\frac{-22±4\sqrt{31}}{6}$$

t=\frac{-22±4\sqrt{31}}{6} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=\frac{2\sqrt{31}-11}{3} t=\frac{-2\sqrt{31}-11}{3}$$

x=t^{2} boʻlganda, yechimlar musbat t uchun x=±\sqrt{t} hisoblanishi orqali olinadi.

$$x=\sqrt{\frac{2\sqrt{31}-11}{3}} x=-\sqrt{\frac{2\sqrt{31}-11}{3}}$$