frac{6(k^{2}+1)^{2}-(3k^2-1)^2}{(3k^2+1)^2}=5/4 eching

$$\frac { 6 ( k ^ { 2 } + 1 ) ^ { 2 } - ( 3 k ^ { 2 } - 1 ) ^ { 2 } } { ( 3 k ^ { 2 } + 1 ) ^ { 2 } } = \frac { 5 } { 4 }$$

k uchun yechish

$$k=-1$$

$$k=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 4\left(3k^{2}+1\right)^{2} ga, \left(3k^{2}+1\right)^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$4\left(6\left(k^{2}+1\right)^{2}-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

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

$$4\left(6\left(\left(k^{2}\right)^{2}+2k^{2}+1\right)-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

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

$$4\left(6\left(k^{4}+2k^{2}+1\right)-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

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

$$4\left(6k^{4}+12k^{2}+6-\left(3k^{2}-1\right)^{2}\right)=5\left(3k^{2}+1\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3k^{2}-1\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(6k^{4}+12k^{2}+6-\left(9\left(k^{2}\right)^{2}-6k^{2}+1\right)\right)=5\left(3k^{2}+1\right)^{2}$$

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

$$4\left(6k^{4}+12k^{2}+6-\left(9k^{4}-6k^{2}+1\right)\right)=5\left(3k^{2}+1\right)^{2}$$

9k^{4}-6k^{2}+1 teskarisini topish uchun har birining teskarisini toping.

$$4\left(6k^{4}+12k^{2}+6-9k^{4}+6k^{2}-1\right)=5\left(3k^{2}+1\right)^{2}$$

-3k^{4} ni olish uchun 6k^{4} va -9k^{4} ni birlashtirish.

$$4\left(-3k^{4}+12k^{2}+6+6k^{2}-1\right)=5\left(3k^{2}+1\right)^{2}$$

18k^{2} ni olish uchun 12k^{2} va 6k^{2} ni birlashtirish.

$$4\left(-3k^{4}+18k^{2}+6-1\right)=5\left(3k^{2}+1\right)^{2}$$

5 olish uchun 6 dan 1 ni ayirish.

$$4\left(-3k^{4}+18k^{2}+5\right)=5\left(3k^{2}+1\right)^{2}$$

4 ga -3k^{4}+18k^{2}+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-12k^{4}+72k^{2}+20=5\left(3k^{2}+1\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3k^{2}+1\right)^{2} kengaytirilishi uchun ishlating.

$$-12k^{4}+72k^{2}+20=5\left(9\left(k^{2}\right)^{2}+6k^{2}+1\right)$$

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

$$-12k^{4}+72k^{2}+20=5\left(9k^{4}+6k^{2}+1\right)$$

5 ga 9k^{4}+6k^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-12k^{4}+72k^{2}+20=45k^{4}+30k^{2}+5$$

Ikkala tarafdan 45k^{4} ni ayirish.

$$-12k^{4}+72k^{2}+20-45k^{4}=30k^{2}+5$$

-57k^{4} ni olish uchun -12k^{4} va -45k^{4} ni birlashtirish.

$$-57k^{4}+72k^{2}+20=30k^{2}+5$$

Ikkala tarafdan 30k^{2} ni ayirish.

$$-57k^{4}+72k^{2}+20-30k^{2}=5$$

42k^{2} ni olish uchun 72k^{2} va -30k^{2} ni birlashtirish.

$$-57k^{4}+42k^{2}+20=5$$

Ikkala tarafdan 5 ni ayirish.

$$-57k^{4}+42k^{2}+20-5=0$$

15 olish uchun 20 dan 5 ni ayirish.

$$-57k^{4}+42k^{2}+15=0$$

k^{2} uchun t ni almashtiring.

$$-57t^{2}+42t+15=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun -57 ni, b uchun 42 ni va c uchun 15 ni ayiring.

$$t=\frac{-42±\sqrt{42^{2}-4\left(-57\right)\times 15}}{-57\times 2}$$

Hisoblarni amalga oshiring.

$$t=\frac{-42±72}{-114}$$

t=\frac{-42±72}{-114} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$t=-\frac{5}{19} t=1$$

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

$$k=1 k=-1$$