3x^2-6x-1=0 eching

$$3 { x }^{ 2 } -6x-1=0$$

x uchun yechish

$$x=\frac{2\sqrt{3}}{3}+1\approx 2,154700538$$

$$x=-\frac{2\sqrt{3}}{3}+1\approx -0,154700538$$

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.

$$3x^{2}-6x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -6 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-1\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\left(-1\right)}}{2\times 3}$$

-12 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+12}}{2\times 3}$$

36 ni 12 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{48}}{2\times 3}$$

48 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±4\sqrt{3}}{2\times 3}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±4\sqrt{3}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{6±4\sqrt{3}}{6}$$

x=\frac{6±4\sqrt{3}}{6} tenglamasini yeching, bunda ± musbat. 6 ni 4\sqrt{3} ga qo'shish.

$$x=\frac{4\sqrt{3}+6}{6}$$

6+4\sqrt{3} ni 6 ga bo'lish.

$$x=\frac{2\sqrt{3}}{3}+1$$

x=\frac{6±4\sqrt{3}}{6} tenglamasini yeching, bunda ± manfiy. 6 dan 4\sqrt{3} ni ayirish.

$$x=\frac{6-4\sqrt{3}}{6}$$

6-4\sqrt{3} ni 6 ga bo'lish.

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

Tenglama yechildi.

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

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.

$$3x^{2}-6x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-6x-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$3x^{2}-6x=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$3x^{2}-6x=1$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-6x}{3}=\frac{1}{3}$$

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

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{1}{3}$$

-6 ni 3 ga bo'lish.

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

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

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

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

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

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-1=\frac{2\sqrt{3}}{3} x-1=-\frac{2\sqrt{3}}{3}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

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