left(2xright)^2+5x+6=0 eching

$${ \left(2x \right) }^{ 2 } +5x+6=0$$

x uchun yechish (complex solution)

$$x=\frac{-5+\sqrt{71}i}{8}\approx -0,625+1,053268722i$$

$$x=\frac{-\sqrt{71}i-5}{8}\approx -0,625-1,053268722i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(2x\right)^{2} ni kengaytirish.

$$2^{2}x^{2}+5x+6=0$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4x^{2}+5x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 5 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\times 4\times 6}}{2\times 4}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 4\times 6}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-16\times 6}}{2\times 4}$$

-16 ni 6 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-96}}{2\times 4}$$

25 ni -96 ga qo'shish.

$$x=\frac{-5±\sqrt{-71}}{2\times 4}$$

-71 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±\sqrt{71}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{71}i}{8}$$

x=\frac{-5±\sqrt{71}i}{8} tenglamasini yeching, bunda ± musbat. -5 ni i\sqrt{71} ga qo'shish.

$$x=\frac{-5+\sqrt{71}i}{8}$$

x=\frac{-5±\sqrt{71}i}{8} tenglamasini yeching, bunda ± manfiy. -5 dan i\sqrt{71} ni ayirish.

$$x=\frac{-\sqrt{71}i-5}{8}$$

Tenglama yechildi.

$$x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(2x\right)^{2} ni kengaytirish.

$$2^{2}x^{2}+5x+6=0$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4x^{2}+5x+6=0$$

Ikkala tarafdan 6 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$4x^{2}+5x=-6$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+5x}{4}=-\frac{6}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{5}{4}x=-\frac{6}{4}$$

\frac{-6}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{5}{4}x=-\frac{3}{2}$$

\frac{5}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{8} olish uchun. Keyin, \frac{5}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{3}{2}+\left(\frac{5}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{8} kvadratini chiqarish.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{3}{2}+\frac{25}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{2} ni \frac{25}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{71}{64}$$

x^{2}+\frac{5}{4}x+\frac{25}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{8}\right)^{2}=-\frac{71}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{-\frac{71}{64}}$$

Qisqartirish.

$$x+\frac{5}{8}=\frac{\sqrt{71}i}{8} x+\frac{5}{8}=-\frac{\sqrt{71}i}{8}$$

Tenglamaning ikkala tarafidan \frac{5}{8} ni ayirish.

$$x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}$$