Solve (k+2)x^2-2(k-1)x+3=0

$$( k + 2 ) x ^ { 2 } - 2 ( k - 1 ) x + 3 = 0$$

Solve for k

$$k=-\frac{2x^{2}+2x+3}{x\left(x-2\right)}$$

$$x\neq 2\text{ and }x\neq 0$$

Steps for Solving Linear Equation

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

$$kx^{2}+2x^{2}-2\left(k-1\right)x+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$kx^{2}+2x^{2}-2\left(k-1\right)x=-3$$

Use the distributive property to multiply -2 by k-1.

$$kx^{2}+2x^{2}+\left(-2k+2\right)x=-3$$

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

$$kx^{2}+2x^{2}-2kx+2x=-3$$

Subtract 2x^{2} from both sides.

$$kx^{2}-2kx+2x=-3-2x^{2}$$

Subtract 2x from both sides.

$$kx^{2}-2kx=-3-2x^{2}-2x$$

Combine all terms containing k.

$$\left(x^{2}-2x\right)k=-3-2x^{2}-2x$$

The equation is in standard form.

$$\left(x^{2}-2x\right)k=-2x^{2}-2x-3$$

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

$$\frac{\left(x^{2}-2x\right)k}{x^{2}-2x}=\frac{-2x^{2}-2x-3}{x^{2}-2x}$$

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

$$k=\frac{-2x^{2}-2x-3}{x^{2}-2x}$$

Divide -3-2x^{2}-2x by x^{2}-2x.

$$k=-\frac{2x^{2}+2x+3}{x\left(x-2\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{k^{2}-5k-5}+k-1}{k+2}\text{; }x=\frac{-\sqrt{k^{2}-5k-5}+k-1}{k+2}\text{, }&k\neq -2\\x=-\frac{1}{2}\text{, }&k=-2\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{k^{2}-5k-5}+k-1}{k+2}\text{; }x=\frac{-\sqrt{k^{2}-5k-5}+k-1}{k+2}\text{, }&\left(k\neq -2\text{ and }k\leq \frac{5-3\sqrt{5}}{2}\right)\text{ or }k\geq \frac{3\sqrt{5}+5}{2}\\x=-\frac{1}{2}\text{, }&k=-2\end{matrix}\right.$$