219x^2-12x+4=0 eching

$$219 x ^ { 2 } - 12 x + 4 = 0$$

x uchun yechish (complex solution)

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73}\approx 0,02739726+0,13234134i$$

$$x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}\approx 0,02739726-0,13234134i$$

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.

$$219x^{2}-12x+4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 219 ni a, -12 ni b va 4 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 219\times 4}}{2\times 219}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 219\times 4}}{2\times 219}$$

-4 ni 219 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-876\times 4}}{2\times 219}$$

-876 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-3504}}{2\times 219}$$

144 ni -3504 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{-3360}}{2\times 219}$$

-3360 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{210}i}{2\times 219}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{210}i}{2\times 219}$$

2 ni 219 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{210}i}{438}$$

x=\frac{12±4\sqrt{210}i}{438} tenglamasini yeching, bunda ± musbat. 12 ni 4i\sqrt{210} ga qo'shish.

$$x=\frac{12+4\sqrt{210}i}{438}$$

12+4i\sqrt{210} ni 438 ga bo'lish.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

x=\frac{12±4\sqrt{210}i}{438} tenglamasini yeching, bunda ± manfiy. 12 dan 4i\sqrt{210} ni ayirish.

$$x=\frac{-4\sqrt{210}i+12}{438}$$

12-4i\sqrt{210} ni 438 ga bo'lish.

$$x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73} x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

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.

$$219x^{2}-12x+4=0$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$219x^{2}-12x+4-4=-4$$

O‘zidan 4 ayirilsa 0 qoladi.

$$219x^{2}-12x=-4$$

Ikki tarafini 219 ga bo‘ling.

$$\frac{219x^{2}-12x}{219}=-\frac{4}{219}$$

219 ga bo'lish 219 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{219}\right)x=-\frac{4}{219}$$

\frac{-12}{219} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{4}{73}x=-\frac{4}{219}$$

-\frac{4}{73} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{2}{73} olish uchun. Keyin, -\frac{2}{73} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{4}{73}x+\left(-\frac{2}{73}\right)^{2}=-\frac{4}{219}+\left(-\frac{2}{73}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{73} kvadratini chiqarish.

$$x^{2}-\frac{4}{73}x+\frac{4}{5329}=-\frac{4}{219}+\frac{4}{5329}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{219} ni \frac{4}{5329} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{4}{73}x+\frac{4}{5329}=-\frac{280}{15987}$$

x^{2}-\frac{4}{73}x+\frac{4}{5329} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{2}{73}\right)^{2}=-\frac{280}{15987}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{73}\right)^{2}}=\sqrt{-\frac{280}{15987}}$$

Qisqartirish.

$$x-\frac{2}{73}=\frac{2\sqrt{210}i}{219} x-\frac{2}{73}=-\frac{2\sqrt{210}i}{219}$$

\frac{2}{73} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73} x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$