2x^2-4x+12=0 eching

$$2 x ^ { 2 } - 4 x + 12 = 0$$

x uchun yechish (complex solution)

$$x=1+\sqrt{5}i\approx 1+2,236067977i$$

$$x=-\sqrt{5}i+1\approx 1-2,236067977i$$

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.

$$2x^{2}-4x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -4 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 12}}{2\times 2}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\times 12}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\times 12}}{2\times 2}$$

-8 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-96}}{2\times 2}$$

16 ni -96 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-80}}{2\times 2}$$

-80 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±4\sqrt{5}i}{2\times 2}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±4\sqrt{5}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{4±4\sqrt{5}i}{4}$$

x=\frac{4±4\sqrt{5}i}{4} tenglamasini yeching, bunda ± musbat. 4 ni 4i\sqrt{5} ga qo'shish.

$$x=\frac{4+4\sqrt{5}i}{4}$$

4+4i\sqrt{5} ni 4 ga bo'lish.

$$x=1+\sqrt{5}i$$

x=\frac{4±4\sqrt{5}i}{4} tenglamasini yeching, bunda ± manfiy. 4 dan 4i\sqrt{5} ni ayirish.

$$x=\frac{-4\sqrt{5}i+4}{4}$$

4-4i\sqrt{5} ni 4 ga bo'lish.

$$x=-\sqrt{5}i+1$$

Tenglama yechildi.

$$x=1+\sqrt{5}i x=-\sqrt{5}i+1$$

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.

$$2x^{2}-4x+12=0$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$2x^{2}-4x+12-12=-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$2x^{2}-4x=-12$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-4x}{2}=-\frac{12}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-4 ni 2 ga bo'lish.

$$x^{2}-2x=-\frac{12}{2}$$

-12 ni 2 ga bo'lish.

$$x^{2}-2x=-6$$

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=-6+1$$

-6 ni 1 ga qo'shish.

$$x^{2}-2x+1=-5$$

x^{2}-2x+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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-1=\sqrt{5}i x-1=-\sqrt{5}i$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=1+\sqrt{5}i x=-\sqrt{5}i+1$$