2x^2+6x^2+7x+60=0 eching

$$2 { x }^{ 2 } +6 { x }^{ 2 } +7x+60 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-7+\sqrt{1871}i}{16}\approx -0,4375+2,703441094i$$

$$x=\frac{-\sqrt{1871}i-7}{16}\approx -0,4375-2,703441094i$$

Kvadrat tenglamalaridan foydalanish qoidalari

8x^{2} ni olish uchun 2x^{2} va 6x^{2} ni birlashtirish.

$$8x^{2}+7x+60=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 7 ni b va 60 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 8\times 60}}{2\times 8}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 8\times 60}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-32\times 60}}{2\times 8}$$

-32 ni 60 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-1920}}{2\times 8}$$

49 ni -1920 ga qo'shish.

$$x=\frac{-7±\sqrt{-1871}}{2\times 8}$$

-1871 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±\sqrt{1871}i}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{1871}i}{16}$$

x=\frac{-7±\sqrt{1871}i}{16} tenglamasini yeching, bunda ± musbat. -7 ni i\sqrt{1871} ga qo'shish.

$$x=\frac{-7+\sqrt{1871}i}{16}$$

x=\frac{-7±\sqrt{1871}i}{16} tenglamasini yeching, bunda ± manfiy. -7 dan i\sqrt{1871} ni ayirish.

$$x=\frac{-\sqrt{1871}i-7}{16}$$

Tenglama yechildi.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

8x^{2} ni olish uchun 2x^{2} va 6x^{2} ni birlashtirish.

$$8x^{2}+7x+60=0$$

Ikkala tarafdan 60 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$8x^{2}+7x=-60$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}+7x}{8}=-\frac{60}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{8}x=-\frac{60}{8}$$

\frac{-60}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{7}{8}x=-\frac{15}{2}$$

\frac{7}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{16} olish uchun. Keyin, \frac{7}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{8}x+\left(\frac{7}{16}\right)^{2}=-\frac{15}{2}+\left(\frac{7}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{16} kvadratini chiqarish.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{15}{2}+\frac{49}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{15}{2} ni \frac{49}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{8}x+\frac{49}{256}=-\frac{1871}{256}$$

x^{2}+\frac{7}{8}x+\frac{49}{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.

$$\left(x+\frac{7}{16}\right)^{2}=-\frac{1871}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{16}\right)^{2}}=\sqrt{-\frac{1871}{256}}$$

Qisqartirish.

$$x+\frac{7}{16}=\frac{\sqrt{1871}i}{16} x+\frac{7}{16}=-\frac{\sqrt{1871}i}{16}$$

Tenglamaning ikkala tarafidan \frac{7}{16} ni ayirish.

$$x=\frac{-7+\sqrt{1871}i}{16} x=\frac{-\sqrt{1871}i-7}{16}$$