Solve kx^2-(2k+1)x+k+1=0

$$k x ^ { 2 } - ( 2 k + 1 ) x + k + 1 = 0$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{1}{x-1}\text{, }&x\neq 1\\k\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2k+1 by x.

$$kx^{2}-\left(2kx+x\right)+k+1=0$$

To find the opposite of 2kx+x, find the opposite of each term.

$$kx^{2}-2kx-x+k+1=0$$

Add x to both sides. Anything plus zero gives itself.

$$kx^{2}-2kx+k+1=x$$

Subtract 1 from both sides.

$$kx^{2}-2kx+k=x-1$$

Combine all terms containing k.

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

Divide both sides by x^{2}-2x+1.

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

Dividing by x^{2}-2x+1 undoes the multiplication by x^{2}-2x+1.

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

Divide -1+x by x^{2}-2x+1.

$$k=\frac{1}{x-1}$$

Solve for k

$$\left\{\begin{matrix}k=\frac{1}{x-1}\text{, }&x\neq 1\\k\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2k+1 by x.

$$kx^{2}-\left(2kx+x\right)+k+1=0$$

To find the opposite of 2kx+x, find the opposite of each term.

$$kx^{2}-2kx-x+k+1=0$$

Add x to both sides. Anything plus zero gives itself.

$$kx^{2}-2kx+k+1=x$$

Subtract 1 from both sides.

$$kx^{2}-2kx+k=x-1$$

Combine all terms containing k.

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

Divide both sides by x^{2}-2x+1.

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

Dividing by x^{2}-2x+1 undoes the multiplication by x^{2}-2x+1.

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

Divide -1+x by x^{2}-2x+1.

$$k=\frac{1}{x-1}$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=1+\frac{1}{k}\text{, }&k\neq 0\end{matrix}\right.$$