2x^2-4x+7=0 eching

$$2 { x }^{ 2 } -4x+7=0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{10}i}{2}+1\approx 1+1,58113883i$$

$$x=-\frac{\sqrt{10}i}{2}+1\approx 1-1,58113883i$$

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.

$$2x^{2}-4x+7=0$$

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 7 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 7}}{2\times 2}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\times 7}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\times 7}}{2\times 2}$$

-8 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-56}}{2\times 2}$$

16 ni -56 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-40}}{2\times 2}$$

-40 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±2\sqrt{10}i}{2\times 2}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±2\sqrt{10}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{4±2\sqrt{10}i}{4}$$

x=\frac{4±2\sqrt{10}i}{4} tenglamasini yeching, bunda ± musbat. 4 ni 2i\sqrt{10} ga qo'shish.

$$x=\frac{4+2\sqrt{10}i}{4}$$

4+2i\sqrt{10} ni 4 ga bo'lish.

$$x=\frac{\sqrt{10}i}{2}+1$$

x=\frac{4±2\sqrt{10}i}{4} tenglamasini yeching, bunda ± manfiy. 4 dan 2i\sqrt{10} ni ayirish.

$$x=\frac{-2\sqrt{10}i+4}{4}$$

4-2i\sqrt{10} ni 4 ga bo'lish.

$$x=-\frac{\sqrt{10}i}{2}+1$$

Tenglama yechildi.

$$x=\frac{\sqrt{10}i}{2}+1 x=-\frac{\sqrt{10}i}{2}+1$$

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.

$$2x^{2}-4x+7=0$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$2x^{2}-4x+7-7=-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$2x^{2}-4x=-7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-4x}{2}=-\frac{7}{2}$$

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

$$x^{2}+\left(-\frac{4}{2}\right)x=-\frac{7}{2}$$

-4 ni 2 ga bo'lish.

$$x^{2}-2x=-\frac{7}{2}$$

-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.

$$x^{2}-2x+1=-\frac{7}{2}+1$$

-\frac{7}{2} ni 1 ga qo'shish.

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

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-1=\frac{\sqrt{10}i}{2} x-1=-\frac{\sqrt{10}i}{2}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{10}i}{2}+1 x=-\frac{\sqrt{10}i}{2}+1$$