x^2-4(x^2+x+2)=3x^2+4x+4 eching

$$x ^ { 2 } - 4 ( x ^ { 2 } + x + 2 ) = 3 x ^ { 2 } + 4 x + 4$$

x uchun yechish (complex solution)

$$x=\frac{-2+\sqrt{14}i}{3}\approx -0,666666667+1,247219129i$$

$$x=\frac{-\sqrt{14}i-2}{3}\approx -0,666666667-1,247219129i$$

Kvadrat tenglamalaridan foydalanish qoidalari

-4 ga x^{2}+x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x^{2}-4x-8=3x^{2}+4x+4$$

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$-3x^{2}-4x-8=3x^{2}+4x+4$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$-3x^{2}-4x-8-3x^{2}=4x+4$$

-6x^{2} ni olish uchun -3x^{2} va -3x^{2} ni birlashtirish.

$$-6x^{2}-4x-8=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$-6x^{2}-4x-8-4x=4$$

-8x ni olish uchun -4x va -4x ni birlashtirish.

$$-6x^{2}-8x-8=4$$

Ikkala tarafdan 4 ni ayirish.

$$-6x^{2}-8x-8-4=0$$

-12 olish uchun -8 dan 4 ni ayirish.

$$-6x^{2}-8x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, -8 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+24\left(-12\right)}}{2\left(-6\right)}$$

24 ni -12 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-288}}{2\left(-6\right)}$$

64 ni -288 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{-224}}{2\left(-6\right)}$$

-224 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±4\sqrt{14}i}{2\left(-6\right)}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±4\sqrt{14}i}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{8±4\sqrt{14}i}{-12}$$

x=\frac{8±4\sqrt{14}i}{-12} tenglamasini yeching, bunda ± musbat. 8 ni 4i\sqrt{14} ga qo'shish.

$$x=\frac{8+4\sqrt{14}i}{-12}$$

8+4i\sqrt{14} ni -12 ga bo'lish.

$$x=\frac{-\sqrt{14}i-2}{3}$$

x=\frac{8±4\sqrt{14}i}{-12} tenglamasini yeching, bunda ± manfiy. 8 dan 4i\sqrt{14} ni ayirish.

$$x=\frac{-4\sqrt{14}i+8}{-12}$$

8-4i\sqrt{14} ni -12 ga bo'lish.

$$x=\frac{-2+\sqrt{14}i}{3}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{14}i-2}{3} x=\frac{-2+\sqrt{14}i}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

-4 ga x^{2}+x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x^{2}-4x-8=3x^{2}+4x+4$$

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$-3x^{2}-4x-8=3x^{2}+4x+4$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$-3x^{2}-4x-8-3x^{2}=4x+4$$

-6x^{2} ni olish uchun -3x^{2} va -3x^{2} ni birlashtirish.

$$-6x^{2}-4x-8=4x+4$$

Ikkala tarafdan 4x ni ayirish.

$$-6x^{2}-4x-8-4x=4$$

-8x ni olish uchun -4x va -4x ni birlashtirish.

$$-6x^{2}-8x-8=4$$

8 ni ikki tarafga qo’shing.

$$-6x^{2}-8x=4+8$$

12 olish uchun 4 va 8'ni qo'shing.

$$-6x^{2}-8x=12$$

Ikki tarafini -6 ga bo‘ling.

$$\frac{-6x^{2}-8x}{-6}=\frac{12}{-6}$$

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{8}{-6}\right)x=\frac{12}{-6}$$

\frac{-8}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{4}{3}x=\frac{12}{-6}$$

12 ni -6 ga bo'lish.

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

\frac{4}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{2}{3} olish uchun. Keyin, \frac{2}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{2}{3} kvadratini chiqarish.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-2+\frac{4}{9}$$

-2 ni \frac{4}{9} ga qo'shish.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-\frac{14}{9}$$

x^{2}+\frac{4}{3}x+\frac{4}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{2}{3}\right)^{2}=-\frac{14}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{-\frac{14}{9}}$$

Qisqartirish.

$$x+\frac{2}{3}=\frac{\sqrt{14}i}{3} x+\frac{2}{3}=-\frac{\sqrt{14}i}{3}$$

Tenglamaning ikkala tarafidan \frac{2}{3} ni ayirish.

$$x=\frac{-2+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i-2}{3}$$