left(x+2right)^2-x=left(2xright)^2-left(x-2right)^2 eching

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

x uchun yechish

$$x = \frac{\sqrt{65} - 1}{4} \approx 1,765564437$$

$$x=\frac{-\sqrt{65}-1}{4}\approx -2,265564437$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

3x ni olish uchun 4x va -x ni birlashtirish.

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

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

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

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

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

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

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

x^{2}-4x+4 teskarisini topish uchun har birining teskarisini toping.

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

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

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

Ikkala tarafdan 3x^{2} ni ayirish.

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

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

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

Ikkala tarafdan 4x ni ayirish.

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

-x ni olish uchun 3x va -4x ni birlashtirish.

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

4 ni ikki tarafga qo’shing.

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

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

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, -1 ni b va 8 ni c bilan almashtiring.

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

-4 ni -2 marotabaga ko'paytirish.

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

8 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+64}}{2\left(-2\right)}$$

1 ni 64 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{65}}{2\left(-2\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{65}}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{65}}{-4}$$

x=\frac{1±\sqrt{65}}{-4} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{65} ga qo'shish.

$$x=\frac{\sqrt{65}+1}{-4}$$

1+\sqrt{65} ni -4 ga bo'lish.

$$x=\frac{-\sqrt{65}-1}{4}$$

x=\frac{1±\sqrt{65}}{-4} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{65} ni ayirish.

$$x=\frac{1-\sqrt{65}}{-4}$$

1-\sqrt{65} ni -4 ga bo'lish.

$$x=\frac{\sqrt{65}-1}{4}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{65}-1}{4} x=\frac{\sqrt{65}-1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

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

3x ni olish uchun 4x va -x ni birlashtirish.

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

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

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

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

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

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

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

x^{2}-4x+4 teskarisini topish uchun har birining teskarisini toping.

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

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

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

Ikkala tarafdan 3x^{2} ni ayirish.

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

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

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

Ikkala tarafdan 4x ni ayirish.

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

-x ni olish uchun 3x va -4x ni birlashtirish.

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

Ikkala tarafdan 4 ni ayirish.

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

-8 olish uchun -4 dan 4 ni ayirish.

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

Ikki tarafini -2 ga bo‘ling.

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

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

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

-1 ni -2 ga bo'lish.

$$x^{2}+\frac{1}{2}x=-\frac{8}{-2}$$

-8 ni -2 ga bo'lish.

$$x^{2}+\frac{1}{2}x=4$$

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=4+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=4+\frac{1}{16}$$

4 ni \frac{1}{16} ga qo'shish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{65}{16}$$

x^{2}+\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{65}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{65}{16}}$$

Qisqartirish.

$$x+\frac{1}{4}=\frac{\sqrt{65}}{4} x+\frac{1}{4}=-\frac{\sqrt{65}}{4}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$x=\frac{\sqrt{65}-1}{4} x=\frac{-\sqrt{65}-1}{4}$$