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

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

q uchun yechish

$$q=4px-4x-2p^{2}+3$$

Yechim qadamlari

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

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

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

$$2x^{2}-4x+3=2x^{2}-4xp+2p^{2}+q$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2x^{2}-4xp+2p^{2}+q=2x^{2}-4x+3$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$-4xp+2p^{2}+q=2x^{2}-4x+3-2x^{2}$$

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

$$-4xp+2p^{2}+q=-4x+3$$

4xp ni ikki tarafga qo’shing.

$$2p^{2}+q=-4x+3+4xp$$

Ikkala tarafdan 2p^{2} ni ayirish.

$$q=-4x+3+4xp-2p^{2}$$

p uchun yechish (complex solution)

$$p=\frac{\sqrt{4x^{2}-8x-2q+6}}{2}+x$$

$$p=-\frac{\sqrt{4x^{2}-8x-2q+6}}{2}+x$$

p uchun yechish

$$p=\frac{\sqrt{4x^{2}-8x-2q+6}}{2}+x$$

$$p=-\frac{\sqrt{4x^{2}-8x-2q+6}}{2}+x\text{, }q\leq 2x^{2}-4x+3$$