3x^2+3x+5=0 eching

$$3 { x }^{ 2 } +3x+5=0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{51}i}{6}-\frac{1}{2}\approx -0,5+1,190238071i$$

$$x=-\frac{\sqrt{51}i}{6}-\frac{1}{2}\approx -0,5-1,190238071i$$

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.

$$3x^{2}+3x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 3 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 3\times 5}}{2\times 3}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 3\times 5}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-12\times 5}}{2\times 3}$$

-12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-60}}{2\times 3}$$

9 ni -60 ga qo'shish.

$$x=\frac{-3±\sqrt{-51}}{2\times 3}$$

-51 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±\sqrt{51}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{51}i}{6}$$

x=\frac{-3±\sqrt{51}i}{6} tenglamasini yeching, bunda ± musbat. -3 ni i\sqrt{51} ga qo'shish.

$$x=\frac{-3+\sqrt{51}i}{6}$$

-3+i\sqrt{51} ni 6 ga bo'lish.

$$x=\frac{\sqrt{51}i}{6}-\frac{1}{2}$$

x=\frac{-3±\sqrt{51}i}{6} tenglamasini yeching, bunda ± manfiy. -3 dan i\sqrt{51} ni ayirish.

$$x=\frac{-\sqrt{51}i-3}{6}$$

-3-i\sqrt{51} ni 6 ga bo'lish.

$$x=-\frac{\sqrt{51}i}{6}-\frac{1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{51}i}{6}-\frac{1}{2} x=-\frac{\sqrt{51}i}{6}-\frac{1}{2}$$

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.

$$3x^{2}+3x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$3x^{2}+3x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$3x^{2}+3x=-5$$

Ikki tarafini 3 ga bo‘ling.

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

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

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

3 ni 3 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=-\frac{5}{3}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{3} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+x+\frac{1}{4}=-\frac{17}{12}$$

x^{2}+x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{17}{12}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{17}{12}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{51}i}{6} x+\frac{1}{2}=-\frac{\sqrt{51}i}{6}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{\sqrt{51}i}{6}-\frac{1}{2} x=-\frac{\sqrt{51}i}{6}-\frac{1}{2}$$