left(3xright)^2-4x+1=0 eching

$${ \left(3x \right) }^{ 2 } -4x+1=0$$

x uchun yechish (complex solution)

$$x=\frac{2+\sqrt{5}i}{9}\approx 0,222222222+0,248451997i$$

$$x=\frac{-\sqrt{5}i+2}{9}\approx 0,222222222-0,248451997i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(3x\right)^{2} ni kengaytirish.

$$3^{2}x^{2}-4x+1=0$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$9x^{2}-4x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -4 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 9}}{2\times 9}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 9}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-36}}{2\times 9}$$

16 ni -36 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-20}}{2\times 9}$$

-20 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±2\sqrt{5}i}{2\times 9}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±2\sqrt{5}i}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{4±2\sqrt{5}i}{18}$$

x=\frac{4±2\sqrt{5}i}{18} tenglamasini yeching, bunda ± musbat. 4 ni 2i\sqrt{5} ga qo'shish.

$$x=\frac{4+2\sqrt{5}i}{18}$$

4+2i\sqrt{5} ni 18 ga bo'lish.

$$x=\frac{2+\sqrt{5}i}{9}$$

x=\frac{4±2\sqrt{5}i}{18} tenglamasini yeching, bunda ± manfiy. 4 dan 2i\sqrt{5} ni ayirish.

$$x=\frac{-2\sqrt{5}i+4}{18}$$

4-2i\sqrt{5} ni 18 ga bo'lish.

$$x=\frac{-\sqrt{5}i+2}{9}$$

Tenglama yechildi.

$$x=\frac{2+\sqrt{5}i}{9} x=\frac{-\sqrt{5}i+2}{9}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(3x\right)^{2} ni kengaytirish.

$$3^{2}x^{2}-4x+1=0$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$9x^{2}-4x+1=0$$

Ikkala tarafdan 1 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$9x^{2}-4x=-1$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}-4x}{9}=-\frac{1}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{4}{9}x=-\frac{1}{9}$$

-\frac{4}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{9} olish uchun. Keyin, -\frac{2}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{4}{9}x+\left(-\frac{2}{9}\right)^{2}=-\frac{1}{9}+\left(-\frac{2}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{9} kvadratini chiqarish.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=-\frac{1}{9}+\frac{4}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{9} ni \frac{4}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=-\frac{5}{81}$$

x^{2}-\frac{4}{9}x+\frac{4}{81} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{9}\right)^{2}=-\frac{5}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{9}\right)^{2}}=\sqrt{-\frac{5}{81}}$$

Qisqartirish.

$$x-\frac{2}{9}=\frac{\sqrt{5}i}{9} x-\frac{2}{9}=-\frac{\sqrt{5}i}{9}$$

\frac{2}{9} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2+\sqrt{5}i}{9} x=\frac{-\sqrt{5}i+2}{9}$$