3x^2-4x+8=0 eching

$$3 x ^ { 2 } - 4 x + 8 = 0$$

x uchun yechish (complex solution)

$$x=\frac{2+2\sqrt{5}i}{3}\approx 0,666666667+1,490711985i$$

$$x=\frac{-2\sqrt{5}i+2}{3}\approx 0,666666667-1,490711985i$$

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.

$$3x^{2}-4x+8=0$$

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

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\times 8}}{2\times 3}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\times 8}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\times 8}}{2\times 3}$$

-12 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-96}}{2\times 3}$$

16 ni -96 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-80}}{2\times 3}$$

-80 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±4\sqrt{5}i}{2\times 3}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±4\sqrt{5}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{4±4\sqrt{5}i}{6}$$

x=\frac{4±4\sqrt{5}i}{6} tenglamasini yeching, bunda ± musbat. 4 ni 4i\sqrt{5} ga qo'shish.

$$x=\frac{4+4\sqrt{5}i}{6}$$

4+4i\sqrt{5} ni 6 ga bo'lish.

$$x=\frac{2+2\sqrt{5}i}{3}$$

x=\frac{4±4\sqrt{5}i}{6} tenglamasini yeching, bunda ± manfiy. 4 dan 4i\sqrt{5} ni ayirish.

$$x=\frac{-4\sqrt{5}i+4}{6}$$

4-4i\sqrt{5} ni 6 ga bo'lish.

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

Tenglama yechildi.

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

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.

$$3x^{2}-4x+8=0$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$3x^{2}-4x+8-8=-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$3x^{2}-4x=-8$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-4x}{3}=-\frac{8}{3}$$

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

$$x^{2}-\frac{4}{3}x=-\frac{8}{3}$$

-\frac{4}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{3} olish uchun. Keyin, -\frac{2}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=-\frac{8}{3}+\left(-\frac{2}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{3} kvadratini chiqarish.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{8}{3}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{8}{3} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{20}{9}$$

x^{2}-\frac{4}{3}x+\frac{4}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{3}\right)^{2}=-\frac{20}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{-\frac{20}{9}}$$

Qisqartirish.

$$x-\frac{2}{3}=\frac{2\sqrt{5}i}{3} x-\frac{2}{3}=-\frac{2\sqrt{5}i}{3}$$

\frac{2}{3} ni tenglamaning ikkala tarafiga qo'shish.

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