3x^2-12x+1/3=0 eching

$$3 x ^ { 2 } - 12 x + \frac { 1 } { 3 } = 0$$

x uchun yechish

$$x=\frac{\sqrt{35}}{3}+2\approx 3,972026594$$

$$x=-\frac{\sqrt{35}}{3}+2\approx 0,027973406$$

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.

$$3x^{2}-12x+\frac{1}{3}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -12 ni b va \frac{1}{3} ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times \frac{1}{3}}}{2\times 3}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times \frac{1}{3}}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-12\times \frac{1}{3}}}{2\times 3}$$

-12 ni \frac{1}{3} marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4}}{2\times 3}$$

144 ni -4 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{140}}{2\times 3}$$

140 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±2\sqrt{35}}{2\times 3}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±2\sqrt{35}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{12±2\sqrt{35}}{6}$$

x=\frac{12±2\sqrt{35}}{6} tenglamasini yeching, bunda ± musbat. 12 ni 2\sqrt{35} ga qo'shish.

$$x=\frac{2\sqrt{35}+12}{6}$$

12+2\sqrt{35} ni 6 ga bo'lish.

$$x=\frac{\sqrt{35}}{3}+2$$

x=\frac{12±2\sqrt{35}}{6} tenglamasini yeching, bunda ± manfiy. 12 dan 2\sqrt{35} ni ayirish.

$$x=\frac{12-2\sqrt{35}}{6}$$

12-2\sqrt{35} ni 6 ga bo'lish.

$$x=-\frac{\sqrt{35}}{3}+2$$

Tenglama yechildi.

$$x=\frac{\sqrt{35}}{3}+2 x=-\frac{\sqrt{35}}{3}+2$$

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.

$$3x^{2}-12x+\frac{1}{3}=0$$

Tenglamaning ikkala tarafidan \frac{1}{3} ni ayirish.

$$3x^{2}-12x+\frac{1}{3}-\frac{1}{3}=-\frac{1}{3}$$

O‘zidan \frac{1}{3} ayirilsa 0 qoladi.

$$3x^{2}-12x=-\frac{1}{3}$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-12x}{3}=-\frac{\frac{1}{3}}{3}$$

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

$$x^{2}+\left(-\frac{12}{3}\right)x=-\frac{\frac{1}{3}}{3}$$

-12 ni 3 ga bo'lish.

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

-\frac{1}{3} ni 3 ga bo'lish.

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

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

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

-2 kvadratini chiqarish.

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

-\frac{1}{9} ni 4 ga qo'shish.

$$x^{2}-4x+4=\frac{35}{9}$$

x^{2}-4x+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.

$$\left(x-2\right)^{2}=\frac{35}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{35}{9}}$$

Qisqartirish.

$$x-2=\frac{\sqrt{35}}{3} x-2=-\frac{\sqrt{35}}{3}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{35}}{3}+2 x=-\frac{\sqrt{35}}{3}+2$$