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

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

Solve for k

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

$$x\neq -1$$

Steps for Solving Linear Equation

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

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

Subtract x from both sides. Anything subtracted from zero gives its negation.

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

Add 1 to both sides.

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

Combine all terms containing k.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-2k+\sqrt{8k+1}-1}{2k}\text{; }x=-\frac{2k+\sqrt{8k+1}+1}{2k}\text{, }&k\neq 0\\x=1\text{, }&k=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-2k+\sqrt{8k+1}-1}{2k}\text{; }x=-\frac{2k+\sqrt{8k+1}+1}{2k}\text{, }&k\neq 0\text{ and }k\geq -\frac{1}{8}\\x=1\text{, }&k=0\end{matrix}\right.$$