29x^2+8x+7=0 eching

$$29 x ^ { 2 } + 8 x + 7 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-4+\sqrt{187}i}{29}\approx -0,137931034+0,471544632i$$

$$x=\frac{-\sqrt{187}i-4}{29}\approx -0,137931034-0,471544632i$$

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.

$$29x^{2}+8x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 29 ni a, 8 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\times 29\times 7}}{2\times 29}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\times 29\times 7}}{2\times 29}$$

-4 ni 29 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64-116\times 7}}{2\times 29}$$

-116 ni 7 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64-812}}{2\times 29}$$

64 ni -812 ga qo'shish.

$$x=\frac{-8±\sqrt{-748}}{2\times 29}$$

-748 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±2\sqrt{187}i}{2\times 29}$$

2 ni 29 marotabaga ko'paytirish.

$$x=\frac{-8±2\sqrt{187}i}{58}$$

x=\frac{-8±2\sqrt{187}i}{58} tenglamasini yeching, bunda ± musbat. -8 ni 2i\sqrt{187} ga qo'shish.

$$x=\frac{-8+2\sqrt{187}i}{58}$$

-8+2i\sqrt{187} ni 58 ga bo'lish.

$$x=\frac{-4+\sqrt{187}i}{29}$$

x=\frac{-8±2\sqrt{187}i}{58} tenglamasini yeching, bunda ± manfiy. -8 dan 2i\sqrt{187} ni ayirish.

$$x=\frac{-2\sqrt{187}i-8}{58}$$

-8-2i\sqrt{187} ni 58 ga bo'lish.

$$x=\frac{-\sqrt{187}i-4}{29}$$

Tenglama yechildi.

$$x=\frac{-4+\sqrt{187}i}{29} x=\frac{-\sqrt{187}i-4}{29}$$

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.

$$29x^{2}+8x+7=0$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$29x^{2}+8x+7-7=-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$29x^{2}+8x=-7$$

Ikki tarafini 29 ga bo‘ling.

$$\frac{29x^{2}+8x}{29}=-\frac{7}{29}$$

29 ga bo'lish 29 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{8}{29}x=-\frac{7}{29}$$

\frac{8}{29} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{29} olish uchun. Keyin, \frac{4}{29} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{8}{29}x+\left(\frac{4}{29}\right)^{2}=-\frac{7}{29}+\left(\frac{4}{29}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{29} kvadratini chiqarish.

$$x^{2}+\frac{8}{29}x+\frac{16}{841}=-\frac{7}{29}+\frac{16}{841}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{29} ni \frac{16}{841} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{29}x+\frac{16}{841}=-\frac{187}{841}$$

x^{2}+\frac{8}{29}x+\frac{16}{841} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{4}{29}\right)^{2}=-\frac{187}{841}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{29}\right)^{2}}=\sqrt{-\frac{187}{841}}$$

Qisqartirish.

$$x+\frac{4}{29}=\frac{\sqrt{187}i}{29} x+\frac{4}{29}=-\frac{\sqrt{187}i}{29}$$

Tenglamaning ikkala tarafidan \frac{4}{29} ni ayirish.

$$x=\frac{-4+\sqrt{187}i}{29} x=\frac{-\sqrt{187}i-4}{29}$$