7x^2+4x+1=0 eching

$$7 x ^ { 2 } + 4 x + 1 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-2+\sqrt{3}i}{7}\approx -0,285714286+0,24743583i$$

$$x=\frac{-\sqrt{3}i-2}{7}\approx -0,285714286-0,24743583i$$

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.

$$7x^{2}+4x+1=0$$

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 1 ni c bilan almashtiring.

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

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\times 7}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-28}}{2\times 7}$$

16 ni -28 ga qo'shish.

$$x=\frac{-4±\sqrt{-12}}{2\times 7}$$

-12 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±2\sqrt{3}i}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-4±2\sqrt{3}i}{14}$$

x=\frac{-4±2\sqrt{3}i}{14} tenglamasini yeching, bunda ± musbat. -4 ni 2i\sqrt{3} ga qo'shish.

$$x=\frac{-4+2\sqrt{3}i}{14}$$

-4+2i\sqrt{3} ni 14 ga bo'lish.

$$x=\frac{-2+\sqrt{3}i}{7}$$

x=\frac{-4±2\sqrt{3}i}{14} tenglamasini yeching, bunda ± manfiy. -4 dan 2i\sqrt{3} ni ayirish.

$$x=\frac{-2\sqrt{3}i-4}{14}$$

-4-2i\sqrt{3} ni 14 ga bo'lish.

$$x=\frac{-\sqrt{3}i-2}{7}$$

Tenglama yechildi.

$$x=\frac{-2+\sqrt{3}i}{7} x=\frac{-\sqrt{3}i-2}{7}$$

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.

$$7x^{2}+4x+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$7x^{2}+4x+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$7x^{2}+4x=-1$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}+4x}{7}=-\frac{1}{7}$$

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

$$x^{2}+\frac{4}{7}x=-\frac{1}{7}$$

\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.

$$x^{2}+\frac{4}{7}x+\left(\frac{2}{7}\right)^{2}=-\frac{1}{7}+\left(\frac{2}{7}\right)^{2}$$

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

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-\frac{1}{7}+\frac{4}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{7} ni \frac{4}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-\frac{3}{49}$$

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.

$$\left(x+\frac{2}{7}\right)^{2}=-\frac{3}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{2}{7}\right)^{2}}=\sqrt{-\frac{3}{49}}$$

Qisqartirish.

$$x+\frac{2}{7}=\frac{\sqrt{3}i}{7} x+\frac{2}{7}=-\frac{\sqrt{3}i}{7}$$

Tenglamaning ikkala tarafidan \frac{2}{7} ni ayirish.

$$x=\frac{-2+\sqrt{3}i}{7} x=\frac{-\sqrt{3}i-2}{7}$$