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

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

x uchun yechish (complex solution)

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4}\approx 0,25+0,322748612i$$

$$x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}\approx 0,25-0,322748612i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

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

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

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 6}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-24}}{2\times 6}$$

9 ni -24 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{-15}}{2\times 6}$$

-15 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{15}i}{2\times 6}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{15}i}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{15}i}{12}$$

x=\frac{3±\sqrt{15}i}{12} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{15} ga qo'shish.

$$x=\frac{3+\sqrt{15}i}{12}$$

3+i\sqrt{15} ni 12 ga bo'lish.

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

x=\frac{3±\sqrt{15}i}{12} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{15} ni ayirish.

$$x=\frac{-\sqrt{15}i+3}{12}$$

3-i\sqrt{15} ni 12 ga bo'lish.

$$x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4} x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

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

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

Ikki tarafini 6 ga bo‘ling.

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

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

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

\frac{-3}{6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{1}{6}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{6}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{6} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{5}{48}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{5}{48}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{5}{48}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{\sqrt{15}i}{12} x-\frac{1}{4}=-\frac{\sqrt{15}i}{12}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4} x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}$$