1/4=(2k+2)^2-(1-x)/(8-k)^2 eching

$$\frac { 1 } { 4 } = \frac { ( 2 k + 2 ) ^ { 2 } - ( 1 - x ) } { ( 8 - k ) ^ { 2 } }$$

x uchun yechish

$$x=-\frac{15k^{2}}{4}-12k+13$$

$$k\neq 8$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

Tenglamaning ikkala tarafini 4\left(k-8\right)^{2} ga, 4,\left(8-k\right)^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(k-8\right)^{2}=4\left(\left(2k+2\right)^{2}-\left(1-x\right)\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(k-8\right)^{2} kengaytirilishi uchun ishlating.

$$k^{2}-16k+64=4\left(\left(2k+2\right)^{2}-\left(1-x\right)\right)$$

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

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-\left(1-x\right)\right)$$

1-x teskarisini topish uchun har birining teskarisini toping.

$$k^{2}-16k+64=4\left(4k^{2}+8k+4-1+x\right)$$

3 olish uchun 4 dan 1 ni ayirish.

$$k^{2}-16k+64=4\left(4k^{2}+8k+3+x\right)$$

4 ga 4k^{2}+8k+3+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$k^{2}-16k+64=16k^{2}+32k+12+4x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$16k^{2}+32k+12+4x=k^{2}-16k+64$$

Ikkala tarafdan 16k^{2} ni ayirish.

$$32k+12+4x=k^{2}-16k+64-16k^{2}$$

-15k^{2} ni olish uchun k^{2} va -16k^{2} ni birlashtirish.

$$32k+12+4x=-15k^{2}-16k+64$$

Ikkala tarafdan 32k ni ayirish.

$$12+4x=-15k^{2}-16k+64-32k$$

-48k ni olish uchun -16k va -32k ni birlashtirish.

$$12+4x=-15k^{2}-48k+64$$

Ikkala tarafdan 12 ni ayirish.

$$4x=-15k^{2}-48k+64-12$$

52 olish uchun 64 dan 12 ni ayirish.

$$4x=-15k^{2}-48k+52$$

Tenglama standart shaklda.

$$4x=52-48k-15k^{2}$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x}{4}=\frac{52-48k-15k^{2}}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x=\frac{52-48k-15k^{2}}{4}$$

-15k^{2}-48k+52 ni 4 ga bo'lish.

$$x=-\frac{15k^{2}}{4}-12k+13$$

k uchun yechish (complex solution)

$$\left\{\begin{matrix}\\k=-\frac{2\sqrt{339-15x}}{15}-\frac{8}{5}\text{, }&\text{unconditionally}\\k=\frac{2\sqrt{339-15x}}{15}-\frac{8}{5}\text{, }&x\neq -323\end{matrix}\right,$$

k uchun yechish

$$\left\{\begin{matrix}k=\frac{2\sqrt{339-15x}}{15}-\frac{8}{5}\text{, }&x\neq -323\text{ and }x\leq \frac{113}{5}\\k=-\frac{2\sqrt{339-15x}}{15}-\frac{8}{5}\text{, }&x\leq \frac{113}{5}\end{matrix}\right,$$