64x^2+24sqrt{5}x+33=0 eching

$$64 x ^ { 2 } + 24 \sqrt { 5 } x + 33 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}\approx -0,419262746+0,582961191i$$

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}\approx -0,419262746-0,582961191i$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 64 ni a, 24\sqrt{5} ni b va 33 ni c bilan almashtiring.

$$x=\frac{-24\sqrt{5}±\sqrt{\left(24\sqrt{5}\right)^{2}-4\times 64\times 33}}{2\times 64}$$

24\sqrt{5} kvadratini chiqarish.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-4\times 64\times 33}}{2\times 64}$$

-4 ni 64 marotabaga ko'paytirish.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-256\times 33}}{2\times 64}$$

-256 ni 33 marotabaga ko'paytirish.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-8448}}{2\times 64}$$

2880 ni -8448 ga qo'shish.

$$x=\frac{-24\sqrt{5}±\sqrt{-5568}}{2\times 64}$$

-5568 ning kvadrat ildizini chiqarish.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{2\times 64}$$

2 ni 64 marotabaga ko'paytirish.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128}$$

x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} tenglamasini yeching, bunda ± musbat. -24\sqrt{5} ni 8i\sqrt{87} ga qo'shish.

$$x=\frac{-24\sqrt{5}+8\sqrt{87}i}{128}$$

-24\sqrt{5}+8i\sqrt{87} ni 128 ga bo'lish.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}$$

x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} tenglamasini yeching, bunda ± manfiy. -24\sqrt{5} dan 8i\sqrt{87} ni ayirish.

$$x=\frac{-8\sqrt{87}i-24\sqrt{5}}{128}$$

-24\sqrt{5}-8i\sqrt{87} ni 128 ga bo'lish.

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

Tenglamaning ikkala tarafidan 33 ni ayirish.

$$64x^{2}+24\sqrt{5}x+33-33=-33$$

O‘zidan 33 ayirilsa 0 qoladi.

$$64x^{2}+24\sqrt{5}x=-33$$

Ikki tarafini 64 ga bo‘ling.

$$\frac{64x^{2}+24\sqrt{5}x}{64}=-\frac{33}{64}$$

64 ga bo'lish 64 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{24\sqrt{5}}{64}x=-\frac{33}{64}$$

24\sqrt{5} ni 64 ga bo'lish.

$$x^{2}+\frac{3\sqrt{5}}{8}x=-\frac{33}{64}$$

\frac{3\sqrt{5}}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3\sqrt{5}}{16} olish uchun. Keyin, \frac{3\sqrt{5}}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\left(\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{33}{64}+\left(\frac{3\sqrt{5}}{16}\right)^{2}$$

\frac{3\sqrt{5}}{16} kvadratini chiqarish.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{33}{64}+\frac{45}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{33}{64} ni \frac{45}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{87}{256}$$

x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{87}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3\sqrt{5}}{16}\right)^{2}}=\sqrt{-\frac{87}{256}}$$

Qisqartirish.

$$x+\frac{3\sqrt{5}}{16}=\frac{\sqrt{87}i}{16} x+\frac{3\sqrt{5}}{16}=-\frac{\sqrt{87}i}{16}$$

Tenglamaning ikkala tarafidan \frac{3\sqrt{5}}{16} ni ayirish.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$