28k^2+k+1=0 eching

$$28 k ^ { 2 } + k + 1 = 0$$

k uchun yechish

$$k=\frac{-1+\sqrt{111}i}{56}\approx -0,017857143+0,188136674i$$

$$k=\frac{-\sqrt{111}i-1}{56}\approx -0,017857143-0,188136674i$$

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.

$$28k^{2}+k+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 28 ni a, 1 ni b va 1 ni c bilan almashtiring.

$$k=\frac{-1±\sqrt{1^{2}-4\times 28}}{2\times 28}$$

1 kvadratini chiqarish.

$$k=\frac{-1±\sqrt{1-4\times 28}}{2\times 28}$$

-4 ni 28 marotabaga ko'paytirish.

$$k=\frac{-1±\sqrt{1-112}}{2\times 28}$$

1 ni -112 ga qo'shish.

$$k=\frac{-1±\sqrt{-111}}{2\times 28}$$

-111 ning kvadrat ildizini chiqarish.

$$k=\frac{-1±\sqrt{111}i}{2\times 28}$$

2 ni 28 marotabaga ko'paytirish.

$$k=\frac{-1±\sqrt{111}i}{56}$$

k=\frac{-1±\sqrt{111}i}{56} tenglamasini yeching, bunda ± musbat. -1 ni i\sqrt{111} ga qo'shish.

$$k=\frac{-1+\sqrt{111}i}{56}$$

k=\frac{-1±\sqrt{111}i}{56} tenglamasini yeching, bunda ± manfiy. -1 dan i\sqrt{111} ni ayirish.

$$k=\frac{-\sqrt{111}i-1}{56}$$

Tenglama yechildi.

$$k=\frac{-1+\sqrt{111}i}{56} k=\frac{-\sqrt{111}i-1}{56}$$

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.

$$28k^{2}+k+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$28k^{2}+k+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$28k^{2}+k=-1$$

Ikki tarafini 28 ga bo‘ling.

$$\frac{28k^{2}+k}{28}=-\frac{1}{28}$$

28 ga bo'lish 28 ga ko'paytirishni bekor qiladi.

$$k^{2}+\frac{1}{28}k=-\frac{1}{28}$$

\frac{1}{28} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{56} olish uchun. Keyin, \frac{1}{56} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{56} kvadratini chiqarish.

$$k^{2}+\frac{1}{28}k+\frac{1}{3136}=-\frac{1}{28}+\frac{1}{3136}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{28} ni \frac{1}{3136} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}+\frac{1}{28}k+\frac{1}{3136}=-\frac{111}{3136}$$

k^{2}+\frac{1}{28}k+\frac{1}{3136} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k+\frac{1}{56}\right)^{2}=-\frac{111}{3136}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k+\frac{1}{56}\right)^{2}}=\sqrt{-\frac{111}{3136}}$$

Qisqartirish.

$$k+\frac{1}{56}=\frac{\sqrt{111}i}{56} k+\frac{1}{56}=-\frac{\sqrt{111}i}{56}$$

Tenglamaning ikkala tarafidan \frac{1}{56} ni ayirish.

$$k=\frac{-1+\sqrt{111}i}{56} k=\frac{-\sqrt{111}i-1}{56}$$