84x^2+4sqrt{3}x+3=0 eching

$$84 x ^ { 2 } + 4 \sqrt { 3 } x + 3 = 0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}\approx -0,041239305+0,184427778i$$

$$x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}\approx -0,041239305-0,184427778i$$

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.

$$84x^{2}+4\sqrt{3}x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 84 ni a, 4\sqrt{3} ni b va 3 ni c bilan almashtiring.

$$x=\frac{-4\sqrt{3}±\sqrt{\left(4\sqrt{3}\right)^{2}-4\times 84\times 3}}{2\times 84}$$

4\sqrt{3} kvadratini chiqarish.

$$x=\frac{-4\sqrt{3}±\sqrt{48-4\times 84\times 3}}{2\times 84}$$

-4 ni 84 marotabaga ko'paytirish.

$$x=\frac{-4\sqrt{3}±\sqrt{48-336\times 3}}{2\times 84}$$

-336 ni 3 marotabaga ko'paytirish.

$$x=\frac{-4\sqrt{3}±\sqrt{48-1008}}{2\times 84}$$

48 ni -1008 ga qo'shish.

$$x=\frac{-4\sqrt{3}±\sqrt{-960}}{2\times 84}$$

-960 ning kvadrat ildizini chiqarish.

$$x=\frac{-4\sqrt{3}±8\sqrt{15}i}{2\times 84}$$

2 ni 84 marotabaga ko'paytirish.

$$x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168}$$

x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168} tenglamasini yeching, bunda ± musbat. -4\sqrt{3} ni 8i\sqrt{15} ga qo'shish.

$$x=\frac{-4\sqrt{3}+8\sqrt{15}i}{168}$$

-4\sqrt{3}+8i\sqrt{15} ni 168 ga bo'lish.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168} tenglamasini yeching, bunda ± manfiy. -4\sqrt{3} dan 8i\sqrt{15} ni ayirish.

$$x=\frac{-8\sqrt{15}i-4\sqrt{3}}{168}$$

-4\sqrt{3}-8i\sqrt{15} ni 168 ga bo'lish.

$$x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

Tenglama yechildi.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42} x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

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.

$$84x^{2}+4\sqrt{3}x+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$84x^{2}+4\sqrt{3}x+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$84x^{2}+4\sqrt{3}x=-3$$

Ikki tarafini 84 ga bo‘ling.

$$\frac{84x^{2}+4\sqrt{3}x}{84}=-\frac{3}{84}$$

84 ga bo'lish 84 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{4\sqrt{3}}{84}x=-\frac{3}{84}$$

4\sqrt{3} ni 84 ga bo'lish.

$$x^{2}+\frac{\sqrt{3}}{21}x=-\frac{3}{84}$$

\frac{-3}{84} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{\sqrt{3}}{21}x=-\frac{1}{28}$$

\frac{\sqrt{3}}{21} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{\sqrt{3}}{42} olish uchun. Keyin, \frac{\sqrt{3}}{42} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{\sqrt{3}}{21}x+\left(\frac{\sqrt{3}}{42}\right)^{2}=-\frac{1}{28}+\left(\frac{\sqrt{3}}{42}\right)^{2}$$

\frac{\sqrt{3}}{42} kvadratini chiqarish.

$$x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588}=-\frac{1}{28}+\frac{1}{588}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{28} ni \frac{1}{588} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588}=-\frac{5}{147}$$

x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{\sqrt{3}}{42}\right)^{2}=-\frac{5}{147}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{\sqrt{3}}{42}\right)^{2}}=\sqrt{-\frac{5}{147}}$$

Qisqartirish.

$$x+\frac{\sqrt{3}}{42}=\frac{\sqrt{15}i}{21} x+\frac{\sqrt{3}}{42}=-\frac{\sqrt{15}i}{21}$$

Tenglamaning ikkala tarafidan \frac{\sqrt{3}}{42} ni ayirish.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42} x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$