x*(x-1)=-2*(x^2+x+1) eching

$$x \cdot ( x - 1 ) = - 2 \cdot ( x ^ { 2 } + x + 1 )$$

x uchun yechish (complex solution)

$$x=\frac{-1+\sqrt{23}i}{6}\approx -0,166666667+0,799305254i$$

$$x=\frac{-\sqrt{23}i-1}{6}\approx -0,166666667-0,799305254i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-x=-2\left(x^{2}+x+1\right)$$

-2 ga x^{2}+x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-x=-2x^{2}-2x-2$$

2x^{2} ni ikki tarafga qo’shing.

$$x^{2}-x+2x^{2}=-2x-2$$

3x^{2} ni olish uchun x^{2} va 2x^{2} ni birlashtirish.

$$3x^{2}-x=-2x-2$$

2x ni ikki tarafga qo’shing.

$$3x^{2}-x+2x=-2$$

x ni olish uchun -x va 2x ni birlashtirish.

$$3x^{2}+x=-2$$

2 ni ikki tarafga qo’shing.

$$3x^{2}+x+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 1 ni b va 2 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\times 3\times 2}}{2\times 3}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\times 3\times 2}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-12\times 2}}{2\times 3}$$

-12 ni 2 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-24}}{2\times 3}$$

1 ni -24 ga qo'shish.

$$x=\frac{-1±\sqrt{-23}}{2\times 3}$$

-23 ning kvadrat ildizini chiqarish.

$$x=\frac{-1±\sqrt{23}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{23}i}{6}$$

x=\frac{-1±\sqrt{23}i}{6} tenglamasini yeching, bunda ± musbat. -1 ni i\sqrt{23} ga qo'shish.

$$x=\frac{-1+\sqrt{23}i}{6}$$

x=\frac{-1±\sqrt{23}i}{6} tenglamasini yeching, bunda ± manfiy. -1 dan i\sqrt{23} ni ayirish.

$$x=\frac{-\sqrt{23}i-1}{6}$$

Tenglama yechildi.

$$x=\frac{-1+\sqrt{23}i}{6} x=\frac{-\sqrt{23}i-1}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-x=-2\left(x^{2}+x+1\right)$$

-2 ga x^{2}+x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-x=-2x^{2}-2x-2$$

2x^{2} ni ikki tarafga qo’shing.

$$x^{2}-x+2x^{2}=-2x-2$$

3x^{2} ni olish uchun x^{2} va 2x^{2} ni birlashtirish.

$$3x^{2}-x=-2x-2$$

2x ni ikki tarafga qo’shing.

$$3x^{2}-x+2x=-2$$

x ni olish uchun -x va 2x ni birlashtirish.

$$3x^{2}+x=-2$$

Ikki tarafini 3 ga bo‘ling.

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

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1}{3}x=-\frac{2}{3}$$

\frac{1}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{6} olish uchun. Keyin, \frac{1}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{2}{3}+\left(\frac{1}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{6} kvadratini chiqarish.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{2}{3}+\frac{1}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{3} ni \frac{1}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{23}{36}$$

x^{2}+\frac{1}{3}x+\frac{1}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{23}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{23}{36}}$$

Qisqartirish.

$$x+\frac{1}{6}=\frac{\sqrt{23}i}{6} x+\frac{1}{6}=-\frac{\sqrt{23}i}{6}$$

Tenglamaning ikkala tarafidan \frac{1}{6} ni ayirish.

$$x=\frac{-1+\sqrt{23}i}{6} x=\frac{-\sqrt{23}i-1}{6}$$