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

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

Solve for k (complex solution)

$$k=-\frac{2x^{2}+3x-2}{x^{2}+1}$$

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

Steps for Solving Linear Equation

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

$$kx^{2}+2x^{2}+3x+k-2=0$$

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

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

Subtract 3x from both sides.

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

Add 2 to both sides.

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

Combine all terms containing k.

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

The equation is in standard form.

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

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

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

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

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

Solve for k

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

Steps for Solving Linear Equation

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

$$kx^{2}+2x^{2}+3x+k-2=0$$

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

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

Subtract 3x from both sides.

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

Add 2 to both sides.

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

Combine all terms containing k.

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

The equation is in standard form.

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

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

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

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{25-4k^{2}}-3}{2\left(k+2\right)}\text{; }x=-\frac{\sqrt{25-4k^{2}}+3}{2\left(k+2\right)}\text{, }&k\neq -2\\x=\frac{4}{3}\text{, }&k=-2\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{25-4k^{2}}-3}{2\left(k+2\right)}\text{; }x=-\frac{\sqrt{25-4k^{2}}+3}{2\left(k+2\right)}\text{, }&k\neq -2\text{ and }|k|\leq \frac{5}{2}\\x=\frac{4}{3}\text{, }&k=-2\end{matrix}\right.$$