3x^2-4x+12=0 eching

$$3 { x }^{ 2 } -4x+12=0$$

x uchun yechish (complex solution)

$$x=\frac{2+4\sqrt{2}i}{3}\approx 0,666666667+1,885618083i$$

$$x=\frac{-4\sqrt{2}i+2}{3}\approx 0,666666667-1,885618083i$$

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.

$$3x^{2}-4x+12=0$$

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 12 ni c bilan almashtiring.

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

-4 kvadratini chiqarish.

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

-4 ni 3 marotabaga ko'paytirish.

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

-12 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-144}}{2\times 3}$$

16 ni -144 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-128}}{2\times 3}$$

-128 ning kvadrat ildizini chiqarish.

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

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±8\sqrt{2}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{4±8\sqrt{2}i}{6}$$

x=\frac{4±8\sqrt{2}i}{6} tenglamasini yeching, bunda ± musbat. 4 ni 8i\sqrt{2} ga qo'shish.

$$x=\frac{4+8\sqrt{2}i}{6}$$

4+8i\sqrt{2} ni 6 ga bo'lish.

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

x=\frac{4±8\sqrt{2}i}{6} tenglamasini yeching, bunda ± manfiy. 4 dan 8i\sqrt{2} ni ayirish.

$$x=\frac{-8\sqrt{2}i+4}{6}$$

4-8i\sqrt{2} ni 6 ga bo'lish.

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

Tenglama yechildi.

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

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.

$$3x^{2}-4x+12=0$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$3x^{2}-4x+12-12=-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$3x^{2}-4x=-12$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-4x}{3}=-\frac{12}{3}$$

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

$$x^{2}-\frac{4}{3}x=-\frac{12}{3}$$

-12 ni 3 ga bo'lish.

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

-\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.

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

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

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

-4 ni \frac{4}{9} ga qo'shish.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{32}{9}$$

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.

$$\left(x-\frac{2}{3}\right)^{2}=-\frac{32}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{-\frac{32}{9}}$$

Qisqartirish.

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

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

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