x^2+2{left(sqrt{2}div2x-2sqrt{2}right)}^2=8 eching

$${ x }^{ 2 } +2 { \left( \sqrt{ 2 } \div 2x-2 \sqrt{ 2 } \right) }^{ 2 } =8$$

x uchun yechish

$$x=2$$

Guruhlash orqali faktorlash qadamlari

\frac{\sqrt{2}}{2}x ni yagona kasrga aylantiring.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\frac{\sqrt{2}x}{2}-2\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

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

\frac{\sqrt{2}x}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

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

4 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

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

\sqrt{2} kvadrati – 2.

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

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

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

2 ga \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-2\sqrt{2}x\sqrt{2}+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+2\times \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{\left(\sqrt{2}\right)^{2}x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+2\times \frac{2x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{2x^{2}}{4}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\frac{1}{2}x^{2} ni olish uchun 2x^{2} ni 4 ga bo‘ling.

$$x^{2}+2\times \frac{1}{2}x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

1 hosil qilish uchun 2 va \frac{1}{2} ni ko'paytirish.

$$x^{2}+x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+x^{2}-4x\times 2+16=8$$

-8 hosil qilish uchun -4 va 2 ni ko'paytirish.

$$x^{2}+x^{2}-8x+16=8$$

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

$$2x^{2}-8x+16=8$$

Ikkala tarafdan 8 ni ayirish.

$$2x^{2}-8x+16-8=0$$

8 olish uchun 16 dan 8 ni ayirish.

$$2x^{2}-8x+8=0$$

Ikki tarafini 2 ga bo‘ling.

$$x^{2}-4x+4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-4 ab=1\times 4=4$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 4-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-4 -2,-2$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-4=-5 -2-2=-4$$

Yechim – -4 yigʻindisini beruvchi juftlik.

$$a=-2 b=-2$$

x^{2}-4x+4 ni \left(x^{2}-2x\right)+\left(-2x+4\right) sifatida qaytadan yozish.

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

Birinchi guruhda x ni va ikkinchi guruhda -2 ni faktordan chiqaring.

$$x\left(x-2\right)-2\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(x-2\right)$$

Binom kvadrat sifatid qayta yozish.

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

Tenglamani yechish uchun x-2=0 ni yeching.

$$x=2$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{\sqrt{2}}{2}x ni yagona kasrga aylantiring.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\frac{\sqrt{2}x}{2}-2\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

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

\frac{\sqrt{2}x}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

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

4 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

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

\sqrt{2} kvadrati – 2.

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

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

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

2 ga \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-2\sqrt{2}x\sqrt{2}+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+2\times \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{\left(\sqrt{2}\right)^{2}x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+2\times \frac{2x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{2x^{2}}{4}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\frac{1}{2}x^{2} ni olish uchun 2x^{2} ni 4 ga bo‘ling.

$$x^{2}+2\times \frac{1}{2}x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

1 hosil qilish uchun 2 va \frac{1}{2} ni ko'paytirish.

$$x^{2}+x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+x^{2}-4x\times 2+16=8$$

-8 hosil qilish uchun -4 va 2 ni ko'paytirish.

$$x^{2}+x^{2}-8x+16=8$$

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

$$2x^{2}-8x+16=8$$

Ikkala tarafdan 8 ni ayirish.

$$2x^{2}-8x+16-8=0$$

8 olish uchun 16 dan 8 ni ayirish.

$$2x^{2}-8x+8=0$$

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

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

-8 kvadratini chiqarish.

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

-4 ni 2 marotabaga ko'paytirish.

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

-8 ni 8 marotabaga ko'paytirish.

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

64 ni -64 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{0}}{2\times 2}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-8}{2\times 2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{8}{4}$$

8 ni 4 ga bo'lish.

$$x=2$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{\sqrt{2}}{2}x ni yagona kasrga aylantiring.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\frac{\sqrt{2}x}{2}-2\sqrt{2}\right)^{2} kengaytirilishi uchun ishlating.

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

\frac{\sqrt{2}x}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

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

4 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

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

\sqrt{2} kvadrati – 2.

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

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

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

2 ga \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-2\sqrt{2}x\sqrt{2}+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+2\times \frac{\left(\sqrt{2}x\right)^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{\left(\sqrt{2}\right)^{2}x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+2\times \frac{2x^{2}}{2^{2}}-4x\left(\sqrt{2}\right)^{2}+16=8$$

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

$$x^{2}+2\times \frac{2x^{2}}{4}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\frac{1}{2}x^{2} ni olish uchun 2x^{2} ni 4 ga bo‘ling.

$$x^{2}+2\times \frac{1}{2}x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

1 hosil qilish uchun 2 va \frac{1}{2} ni ko'paytirish.

$$x^{2}+x^{2}-4x\left(\sqrt{2}\right)^{2}+16=8$$

\sqrt{2} kvadrati – 2.

$$x^{2}+x^{2}-4x\times 2+16=8$$

-8 hosil qilish uchun -4 va 2 ni ko'paytirish.

$$x^{2}+x^{2}-8x+16=8$$

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

$$2x^{2}-8x+16=8$$

Ikkala tarafdan 16 ni ayirish.

$$2x^{2}-8x=8-16$$

-8 olish uchun 8 dan 16 ni ayirish.

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

Ikki tarafini 2 ga bo‘ling.

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

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-8 ni 2 ga bo'lish.

$$x^{2}-4x=-\frac{8}{2}$$

-8 ni 2 ga bo'lish.

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

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

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

-2 kvadratini chiqarish.

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

-4 ni 4 ga qo'shish.

$$x^{2}-4x+4=0$$

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

$$\left(x-2\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-2=0 x-2=0$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2 x=2$$

Tenglama yechildi. Yechimlar bir xil.

$$x=2$$