3x^2+6x+8=0 eching

$$3 x ^ { 2 } + 6 x + 8 = 0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{15}i}{3}-1\approx -1+1,290994449i$$

$$x=-\frac{\sqrt{15}i}{3}-1\approx -1-1,290994449i$$

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.

$$3x^{2}+6x+8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 6 ni b va 8 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\times 8}}{2\times 3}$$

6 kvadratini chiqarish.

$$x=\frac{-6±\sqrt{36-4\times 3\times 8}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-12\times 8}}{2\times 3}$$

-12 ni 8 marotabaga ko'paytirish.

$$x=\frac{-6±\sqrt{36-96}}{2\times 3}$$

36 ni -96 ga qo'shish.

$$x=\frac{-6±\sqrt{-60}}{2\times 3}$$

-60 ning kvadrat ildizini chiqarish.

$$x=\frac{-6±2\sqrt{15}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-6±2\sqrt{15}i}{6}$$

x=\frac{-6±2\sqrt{15}i}{6} tenglamasini yeching, bunda ± musbat. -6 ni 2i\sqrt{15} ga qo'shish.

$$x=\frac{-6+2\sqrt{15}i}{6}$$

-6+2i\sqrt{15} ni 6 ga bo'lish.

$$x=\frac{\sqrt{15}i}{3}-1$$

x=\frac{-6±2\sqrt{15}i}{6} tenglamasini yeching, bunda ± manfiy. -6 dan 2i\sqrt{15} ni ayirish.

$$x=\frac{-2\sqrt{15}i-6}{6}$$

-6-2i\sqrt{15} ni 6 ga bo'lish.

$$x=-\frac{\sqrt{15}i}{3}-1$$

Tenglama yechildi.

$$x=\frac{\sqrt{15}i}{3}-1 x=-\frac{\sqrt{15}i}{3}-1$$

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.

$$3x^{2}+6x+8=0$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$3x^{2}+6x+8-8=-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$3x^{2}+6x=-8$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+6x}{3}=-\frac{8}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{6}{3}x=-\frac{8}{3}$$

6 ni 3 ga bo'lish.

$$x^{2}+2x=-\frac{8}{3}$$

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.

$$x^{2}+2x+1^{2}=-\frac{8}{3}+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=-\frac{8}{3}+1$$

-\frac{8}{3} ni 1 ga qo'shish.

$$x^{2}+2x+1=-\frac{5}{3}$$

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.

$$\left(x+1\right)^{2}=-\frac{5}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-\frac{5}{3}}$$

Qisqartirish.

$$x+1=\frac{\sqrt{15}i}{3} x+1=-\frac{\sqrt{15}i}{3}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\frac{\sqrt{15}i}{3}-1 x=-\frac{\sqrt{15}i}{3}-1$$