56x^2-12x+1=0 eching

$$56 { x }^{ 2 } -12x+1=0$$

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{5}i}{28}\approx 0,107142857+0,079859571i$$

$$x=\frac{-\sqrt{5}i+3}{28}\approx 0,107142857-0,079859571i$$

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.

$$56x^{2}-12x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 56 ni a, -12 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 56}}{2\times 56}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 56}}{2\times 56}$$

-4 ni 56 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-224}}{2\times 56}$$

144 ni -224 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{-80}}{2\times 56}$$

-80 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4\sqrt{5}i}{2\times 56}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4\sqrt{5}i}{2\times 56}$$

2 ni 56 marotabaga ko'paytirish.

$$x=\frac{12±4\sqrt{5}i}{112}$$

x=\frac{12±4\sqrt{5}i}{112} tenglamasini yeching, bunda ± musbat. 12 ni 4i\sqrt{5} ga qo'shish.

$$x=\frac{12+4\sqrt{5}i}{112}$$

12+4i\sqrt{5} ni 112 ga bo'lish.

$$x=\frac{3+\sqrt{5}i}{28}$$

x=\frac{12±4\sqrt{5}i}{112} tenglamasini yeching, bunda ± manfiy. 12 dan 4i\sqrt{5} ni ayirish.

$$x=\frac{-4\sqrt{5}i+12}{112}$$

12-4i\sqrt{5} ni 112 ga bo'lish.

$$x=\frac{-\sqrt{5}i+3}{28}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{5}i}{28} x=\frac{-\sqrt{5}i+3}{28}$$

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.

$$56x^{2}-12x+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$56x^{2}-12x+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$56x^{2}-12x=-1$$

Ikki tarafini 56 ga bo‘ling.

$$\frac{56x^{2}-12x}{56}=-\frac{1}{56}$$

56 ga bo'lish 56 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{56}\right)x=-\frac{1}{56}$$

\frac{-12}{56} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{14}x=-\frac{1}{56}$$

-\frac{3}{14} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{28} olish uchun. Keyin, -\frac{3}{28} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{14}x+\left(-\frac{3}{28}\right)^{2}=-\frac{1}{56}+\left(-\frac{3}{28}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{28} kvadratini chiqarish.

$$x^{2}-\frac{3}{14}x+\frac{9}{784}=-\frac{1}{56}+\frac{9}{784}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{56} ni \frac{9}{784} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{14}x+\frac{9}{784}=-\frac{5}{784}$$

x^{2}-\frac{3}{14}x+\frac{9}{784} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{28}\right)^{2}=-\frac{5}{784}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{28}\right)^{2}}=\sqrt{-\frac{5}{784}}$$

Qisqartirish.

$$x-\frac{3}{28}=\frac{\sqrt{5}i}{28} x-\frac{3}{28}=-\frac{\sqrt{5}i}{28}$$

\frac{3}{28} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{5}i}{28} x=\frac{-\sqrt{5}i+3}{28}$$