7x^2-4x+6=0 eching

$$7 x ^ { 2 } - 4 x + 6 = 0$$

x uchun yechish (complex solution)

$$x=\frac{2+\sqrt{38}i}{7}\approx 0,285714286+0,880630572i$$

$$x=\frac{-\sqrt{38}i+2}{7}\approx 0,285714286-0,880630572i$$

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.

$$7x^{2}-4x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -4 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 7\times 6}}{2\times 7}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 7\times 6}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-28\times 6}}{2\times 7}$$

-28 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-168}}{2\times 7}$$

16 ni -168 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-152}}{2\times 7}$$

-152 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±2\sqrt{38}i}{2\times 7}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±2\sqrt{38}i}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{4±2\sqrt{38}i}{14}$$

x=\frac{4±2\sqrt{38}i}{14} tenglamasini yeching, bunda ± musbat. 4 ni 2i\sqrt{38} ga qo'shish.

$$x=\frac{4+2\sqrt{38}i}{14}$$

4+2i\sqrt{38} ni 14 ga bo'lish.

$$x=\frac{2+\sqrt{38}i}{7}$$

x=\frac{4±2\sqrt{38}i}{14} tenglamasini yeching, bunda ± manfiy. 4 dan 2i\sqrt{38} ni ayirish.

$$x=\frac{-2\sqrt{38}i+4}{14}$$

4-2i\sqrt{38} ni 14 ga bo'lish.

$$x=\frac{-\sqrt{38}i+2}{7}$$

Tenglama yechildi.

$$x=\frac{2+\sqrt{38}i}{7} x=\frac{-\sqrt{38}i+2}{7}$$

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.

$$7x^{2}-4x+6=0$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$7x^{2}-4x+6-6=-6$$

O‘zidan 6 ayirilsa 0 qoladi.

$$7x^{2}-4x=-6$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-4x}{7}=-\frac{6}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{4}{7}x=-\frac{6}{7}$$

-\frac{4}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{7} olish uchun. Keyin, -\frac{2}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{4}{7}x+\left(-\frac{2}{7}\right)^{2}=-\frac{6}{7}+\left(-\frac{2}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{7} kvadratini chiqarish.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{6}{7}+\frac{4}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{6}{7} ni \frac{4}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{38}{49}$$

x^{2}-\frac{4}{7}x+\frac{4}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{7}\right)^{2}=-\frac{38}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{7}\right)^{2}}=\sqrt{-\frac{38}{49}}$$

Qisqartirish.

$$x-\frac{2}{7}=\frac{\sqrt{38}i}{7} x-\frac{2}{7}=-\frac{\sqrt{38}i}{7}$$

\frac{2}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2+\sqrt{38}i}{7} x=\frac{-\sqrt{38}i+2}{7}$$