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

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

Solve for k (complex solution)

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

$$x\neq -i\text{ and }x\neq i$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-k by x^{2}.

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

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

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

Subtract x from both sides.

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

Subtract 1 from both sides.

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

Combine all terms containing k.

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

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

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

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

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

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

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

Solve for k

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

Steps for Solving Linear Equation

Use the distributive property to multiply 1-k by x^{2}.

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

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

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

Subtract x from both sides.

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

Subtract 1 from both sides.

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

Combine all terms containing k.

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

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

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(3-2k\right)\left(-2\left(1-k\right)+1\right)}-1}{2\left(1-k\right)}\text{; }x=\frac{\sqrt{-4k^{2}+8k-3}+1}{2\left(k-1\right)}\text{, }&k\neq 1\\x=0\text{, }&k=1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{-\left(1-2k\right)\left(3-2k\right)}-1}{2\left(1-k\right)}\text{; }x=-\frac{\sqrt{-\left(1-2k\right)\left(3-2k\right)}+1}{2\left(1-k\right)}\text{, }&k\neq 1\text{ and }k\leq \frac{3}{2}\text{ and }k\geq \frac{1}{2}\\x=0\text{, }&k=1\end{matrix}\right.$$