12x^2-2x+5=0 eching

$$12 x ^ { 2 } - 2 x + 5 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{59}i}{12}\approx 0,083333333+0,640095479i$$

$$x=\frac{-\sqrt{59}i+1}{12}\approx 0,083333333-0,640095479i$$

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.

$$12x^{2}-2x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 12 ni a, -2 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 12\times 5}}{2\times 12}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 12\times 5}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-48\times 5}}{2\times 12}$$

-48 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-240}}{2\times 12}$$

4 ni -240 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{-236}}{2\times 12}$$

-236 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{59}i}{2\times 12}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{59}i}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{59}i}{24}$$

x=\frac{2±2\sqrt{59}i}{24} tenglamasini yeching, bunda ± musbat. 2 ni 2i\sqrt{59} ga qo'shish.

$$x=\frac{2+2\sqrt{59}i}{24}$$

2+2i\sqrt{59} ni 24 ga bo'lish.

$$x=\frac{1+\sqrt{59}i}{12}$$

x=\frac{2±2\sqrt{59}i}{24} tenglamasini yeching, bunda ± manfiy. 2 dan 2i\sqrt{59} ni ayirish.

$$x=\frac{-2\sqrt{59}i+2}{24}$$

2-2i\sqrt{59} ni 24 ga bo'lish.

$$x=\frac{-\sqrt{59}i+1}{12}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{59}i}{12} x=\frac{-\sqrt{59}i+1}{12}$$

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.

$$12x^{2}-2x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$12x^{2}-2x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$12x^{2}-2x=-5$$

Ikki tarafini 12 ga bo‘ling.

$$\frac{12x^{2}-2x}{12}=-\frac{5}{12}$$

12 ga bo'lish 12 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{2}{12}\right)x=-\frac{5}{12}$$

\frac{-2}{12} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{6}x=-\frac{5}{12}$$

-\frac{1}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{12} olish uchun. Keyin, -\frac{1}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=-\frac{5}{12}+\left(-\frac{1}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{12} kvadratini chiqarish.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{5}{12}+\frac{1}{144}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{12} ni \frac{1}{144} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{59}{144}$$

x^{2}-\frac{1}{6}x+\frac{1}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{12}\right)^{2}=-\frac{59}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{-\frac{59}{144}}$$

Qisqartirish.

$$x-\frac{1}{12}=\frac{\sqrt{59}i}{12} x-\frac{1}{12}=-\frac{\sqrt{59}i}{12}$$

\frac{1}{12} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{59}i}{12} x=\frac{-\sqrt{59}i+1}{12}$$