z^4+z^2-12=(z^2+4)^2 eching

$$z ^ { 4 } + z ^ { 2 } - 12 = ( z ^ { 2 } + 4 ) ^ { 2 }$$

z uchun yechish

$$z=-2i$$

$$z=2i$$

Kvadrat ildizini topish bo'yicha qadamlar

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

$$z^{4}+z^{2}-12=\left(z^{2}\right)^{2}+8z^{2}+16$$

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

$$z^{4}+z^{2}-12=z^{4}+8z^{2}+16$$

Ikkala tarafdan z^{4} ni ayirish.

$$z^{4}+z^{2}-12-z^{4}=8z^{2}+16$$

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

$$z^{2}-12=8z^{2}+16$$

Ikkala tarafdan 8z^{2} ni ayirish.

$$z^{2}-12-8z^{2}=16$$

-7z^{2} ni olish uchun z^{2} va -8z^{2} ni birlashtirish.

$$-7z^{2}-12=16$$

12 ni ikki tarafga qo’shing.

$$-7z^{2}=16+12$$

28 olish uchun 16 va 12'ni qo'shing.

$$-7z^{2}=28$$

Ikki tarafini -7 ga bo‘ling.

$$z^{2}=\frac{28}{-7}$$

-4 ni olish uchun 28 ni -7 ga bo‘ling.

$$z^{2}=-4$$

Tenglama yechildi.

$$z=2i z=-2i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$z^{4}+z^{2}-12=\left(z^{2}\right)^{2}+8z^{2}+16$$

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

$$z^{4}+z^{2}-12=z^{4}+8z^{2}+16$$

Ikkala tarafdan z^{4} ni ayirish.

$$z^{4}+z^{2}-12-z^{4}=8z^{2}+16$$

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

$$z^{2}-12=8z^{2}+16$$

Ikkala tarafdan 8z^{2} ni ayirish.

$$z^{2}-12-8z^{2}=16$$

-7z^{2} ni olish uchun z^{2} va -8z^{2} ni birlashtirish.

$$-7z^{2}-12=16$$

Ikkala tarafdan 16 ni ayirish.

$$-7z^{2}-12-16=0$$

-28 olish uchun -12 dan 16 ni ayirish.

$$-7z^{2}-28=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -7 ni a, 0 ni b va -28 ni c bilan almashtiring.

$$z=\frac{0±\sqrt{0^{2}-4\left(-7\right)\left(-28\right)}}{2\left(-7\right)}$$

0 kvadratini chiqarish.

$$z=\frac{0±\sqrt{-4\left(-7\right)\left(-28\right)}}{2\left(-7\right)}$$

-4 ni -7 marotabaga ko'paytirish.

$$z=\frac{0±\sqrt{28\left(-28\right)}}{2\left(-7\right)}$$

28 ni -28 marotabaga ko'paytirish.

$$z=\frac{0±\sqrt{-784}}{2\left(-7\right)}$$

-784 ning kvadrat ildizini chiqarish.

$$z=\frac{0±28i}{2\left(-7\right)}$$

2 ni -7 marotabaga ko'paytirish.

$$z=\frac{0±28i}{-14}$$

z=\frac{0±28i}{-14} tenglamasini yeching, bunda ± musbat.

$$z=-2i$$

z=\frac{0±28i}{-14} tenglamasini yeching, bunda ± manfiy.

$$z=2i$$

Tenglama yechildi.

$$z=-2i z=2i$$