3x^2-2x+9=0 eching

$$3 { x }^{ 2 } -2x+9 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{26}i}{3}\approx 0,333333333+1,699673171i$$

$$x=\frac{-\sqrt{26}i+1}{3}\approx 0,333333333-1,699673171i$$

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.

$$3x^{2}-2x+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -2 ni b va 9 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\times 9}}{2\times 3}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3\times 9}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\times 9}}{2\times 3}$$

-12 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-108}}{2\times 3}$$

4 ni -108 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{-104}}{2\times 3}$$

-104 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{26}i}{2\times 3}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{26}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{26}i}{6}$$

x=\frac{2±2\sqrt{26}i}{6} tenglamasini yeching, bunda ± musbat. 2 ni 2i\sqrt{26} ga qo'shish.

$$x=\frac{2+2\sqrt{26}i}{6}$$

2+2i\sqrt{26} ni 6 ga bo'lish.

$$x=\frac{1+\sqrt{26}i}{3}$$

x=\frac{2±2\sqrt{26}i}{6} tenglamasini yeching, bunda ± manfiy. 2 dan 2i\sqrt{26} ni ayirish.

$$x=\frac{-2\sqrt{26}i+2}{6}$$

2-2i\sqrt{26} ni 6 ga bo'lish.

$$x=\frac{-\sqrt{26}i+1}{3}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{26}i}{3} x=\frac{-\sqrt{26}i+1}{3}$$

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.

$$3x^{2}-2x+9=0$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$3x^{2}-2x+9-9=-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$3x^{2}-2x=-9$$

Ikki tarafini 3 ga bo‘ling.

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

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

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

-9 ni 3 ga bo'lish.

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

-\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{3} olish uchun. Keyin, -\frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-3+\frac{1}{9}$$

-3 ni \frac{1}{9} ga qo'shish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{26}{9}$$

x^{2}-\frac{2}{3}x+\frac{1}{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.

$$\left(x-\frac{1}{3}\right)^{2}=-\frac{26}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{-\frac{26}{9}}$$

Qisqartirish.

$$x-\frac{1}{3}=\frac{\sqrt{26}i}{3} x-\frac{1}{3}=-\frac{\sqrt{26}i}{3}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{26}i}{3} x=\frac{-\sqrt{26}i+1}{3}$$