z^2-2iz+3=0 eching

$$z ^ { 2 } - 2 i z + 3 = 0$$

z uchun yechish

$$z=3i$$

$$z=-i$$

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$z^{2}-2iz+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -2i ni b va 3 ni c bilan almashtiring.

$$z=\frac{2i±\sqrt{\left(-2i\right)^{2}-4\times 3}}{2}$$

-2i kvadratini chiqarish.

$$z=\frac{2i±\sqrt{-4-4\times 3}}{2}$$

-4 ni 3 marotabaga ko'paytirish.

$$z=\frac{2i±\sqrt{-4-12}}{2}$$

-4 ni -12 ga qo'shish.

$$z=\frac{2i±\sqrt{-16}}{2}$$

-16 ning kvadrat ildizini chiqarish.

$$z=\frac{2i±4i}{2}$$

z=\frac{2i±4i}{2} tenglamasini yeching, bunda ± musbat. 2i ni 4i ga qo'shish.

$$z=\frac{6i}{2}$$

6i ni 2 ga bo'lish.

$$z=3i$$

z=\frac{2i±4i}{2} tenglamasini yeching, bunda ± manfiy. 2i dan 4i ni ayirish.

$$z=\frac{-2i}{2}$$

-2i ni 2 ga bo'lish.

$$z=-i$$

Tenglama yechildi.

$$z=3i z=-i$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$z^{2}-2iz+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$z^{2}-2iz+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$z^{2}-2iz=-3$$

-2i ni bo‘lish, x shartining koeffitsienti, 2 ga -i olish uchun. Keyin, -i ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}-2iz+\left(-i\right)^{2}=-3+\left(-i\right)^{2}$$

-i kvadratini chiqarish.

$$z^{2}-2iz-1=-3-1$$

-3 ni -1 ga qo'shish.

$$z^{2}-2iz-1=-4$$

z^{2}-2iz-1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z-i\right)^{2}=-4$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z-i\right)^{2}}=\sqrt{-4}$$

Qisqartirish.

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

i ni tenglamaning ikkala tarafiga qo'shish.

$$z=3i z=-i$$