3(-16k/4k^2+1)^2=32/4k^2+1 eching

$$3 ( \frac { - 16 k } { 4 k ^ { 2 } + 1 } ) ^ { 2 } = \frac { 32 } { 4 k ^ { 2 } + 1 }$$

k uchun yechish

$$k=\frac{\sqrt{5}}{10}\approx 0,223606798$$

$$k=-\frac{\sqrt{5}}{10}\approx -0,223606798$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 4k^{2}+1 ga ko'paytirish.

$$3\times \left(\frac{-16k}{4k^{2}+1}\right)^{2}\left(4k^{2}+1\right)=32$$

\frac{-16k}{4k^{2}+1}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$3\times \frac{\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right)=32$$

3\times \frac{\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}} ni yagona kasrga aylantiring.

$$\frac{3\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right)=32$$

\frac{3\left(-16k\right)^{2}}{\left(4k^{2}+1\right)^{2}}\left(4k^{2}+1\right) ni yagona kasrga aylantiring.

$$\frac{3\left(-16k\right)^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

\left(-16k\right)^{2} ni kengaytirish.

$$\frac{3\left(-16\right)^{2}k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

2 daraja ko‘rsatkichini -16 ga hisoblang va 256 ni qiymatni oling.

$$\frac{3\times 256k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

768 hosil qilish uchun 3 va 256 ni ko'paytirish.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{\left(4k^{2}+1\right)^{2}}=32$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4k^{2}+1\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16\left(k^{2}\right)^{2}+8k^{2}+1}=32$$

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

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16k^{4}+8k^{2}+1}=32$$

Ikkala tarafdan 32 ni ayirish.

$$\frac{768k^{2}\left(4k^{2}+1\right)}{16k^{4}+8k^{2}+1}-32=0$$

768k^{2} ga 4k^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{3072k^{4}+768k^{2}}{16k^{4}+8k^{2}+1}-32=0$$

Faktor: 16k^{4}+8k^{2}+1.

$$\frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}}-32=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 32 ni \frac{\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}} marotabaga ko'paytirish.

$$\frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}}-\frac{32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}=0$$

\frac{3072k^{4}+768k^{2}}{\left(4k^{2}+1\right)^{2}} va \frac{32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{3072k^{4}+768k^{2}-32\left(4k^{2}+1\right)^{2}}{\left(4k^{2}+1\right)^{2}}=0$$

3072k^{4}+768k^{2}-32\left(4k^{2}+1\right)^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{3072k^{4}+768k^{2}-512k^{4}-256k^{2}-32}{\left(4k^{2}+1\right)^{2}}=0$$

3072k^{4}+768k^{2}-512k^{4}-256k^{2}-32 kabi iboralarga o‘xshab birlashtiring.

$$\frac{2560k^{4}+512k^{2}-32}{\left(4k^{2}+1\right)^{2}}=0$$

Tenglamaning ikkala tarafini \left(4k^{2}+1\right)^{2} ga ko'paytirish.

$$2560k^{4}+512k^{2}-32=0$$

k^{2} uchun t ni almashtiring.

$$2560t^{2}+512t-32=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 2560 ni, b uchun 512 ni va c uchun -32 ni ayiring.

$$t=\frac{-512±\sqrt{512^{2}-4\times 2560\left(-32\right)}}{2\times 2560}$$

Hisoblarni amalga oshiring.

$$t=\frac{-512±768}{5120}$$

t=\frac{-512±768}{5120} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=\frac{1}{20} t=-\frac{1}{4}$$

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

$$k=\frac{\sqrt{5}}{10} k=-\frac{\sqrt{5}}{10}$$