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

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

Solve for x

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

$$k\neq 8$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4\left(k-8\right)^{2}, the least common multiple of 4,\left(8-k\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(k-8\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2k+2\right)^{2}.

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

To find the opposite of 1-x, find the opposite of each term.

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

Subtract 1 from 4 to get 3.

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

Use the distributive property to multiply 4 by 4k^{2}+8k+3+x.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 16k^{2} from both sides.

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

Combine k^{2} and -16k^{2} to get -15k^{2}.

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

Subtract 32k from both sides.

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

Combine -16k and -32k to get -48k.

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

Subtract 12 from both sides.

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

Subtract 12 from 64 to get 52.

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

The equation is in standard form.

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

Divide both sides by 4.

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

Dividing by 4 undoes the multiplication by 4.

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

Divide -15k^{2}-48k+52 by 4.

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

Solve for k (complex solution)

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

Solve for k

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