Solve (m-2)x^2-3(m+2)x+6m=0

$$( m - 2 ) x ^ { 2 } - 3 ( m + 2 ) x + 6 m = 0$$

Solve for m (complex solution)

$$m=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

$$x\neq \frac{3+\sqrt{15}i}{2}\text{ and }x\neq \frac{-\sqrt{15}i+3}{2}$$

Steps for Solving Linear Equation

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

$$mx^{2}-2x^{2}-3\left(m+2\right)x+6m=0$$

Use the distributive property to multiply -3 by m+2.

$$mx^{2}-2x^{2}+\left(-3m-6\right)x+6m=0$$

Use the distributive property to multiply -3m-6 by x.

$$mx^{2}-2x^{2}-3mx-6x+6m=0$$

Add 2x^{2} to both sides. Anything plus zero gives itself.

$$mx^{2}-3mx-6x+6m=2x^{2}$$

Add 6x to both sides.

$$mx^{2}-3mx+6m=2x^{2}+6x$$

Combine all terms containing m.

$$\left(x^{2}-3x+6\right)m=2x^{2}+6x$$

Divide both sides by x^{2}-3x+6.

$$\frac{\left(x^{2}-3x+6\right)m}{x^{2}-3x+6}=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

Dividing by x^{2}-3x+6 undoes the multiplication by x^{2}-3x+6.

$$m=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

Solve for m

$$m=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

Steps for Solving Linear Equation

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

$$mx^{2}-2x^{2}-3\left(m+2\right)x+6m=0$$

Use the distributive property to multiply -3 by m+2.

$$mx^{2}-2x^{2}+\left(-3m-6\right)x+6m=0$$

Use the distributive property to multiply -3m-6 by x.

$$mx^{2}-2x^{2}-3mx-6x+6m=0$$

Add 2x^{2} to both sides. Anything plus zero gives itself.

$$mx^{2}-3mx-6x+6m=2x^{2}$$

Add 6x to both sides.

$$mx^{2}-3mx+6m=2x^{2}+6x$$

Combine all terms containing m.

$$\left(x^{2}-3x+6\right)m=2x^{2}+6x$$

Divide both sides by x^{2}-3x+6.

$$\frac{\left(x^{2}-3x+6\right)m}{x^{2}-3x+6}=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

Dividing by x^{2}-3x+6 undoes the multiplication by x^{2}-3x+6.

$$m=\frac{2x\left(x+3\right)}{x^{2}-3x+6}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(12+28m-5m^{2}\right)}+3m+6}{2\left(m-2\right)}\text{; }x=\frac{-\sqrt{3\left(12+28m-5m^{2}\right)}+3m+6}{2\left(m-2\right)}\text{, }&m\neq 2\\x=1\text{, }&m=2\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(12+28m-5m^{2}\right)}+3m+6}{2\left(m-2\right)}\text{; }x=\frac{-\sqrt{3\left(12+28m-5m^{2}\right)}+3m+6}{2\left(m-2\right)}\text{, }&m\neq 2\text{ and }m\leq 6\text{ and }m\geq -\frac{2}{5}\\x=1\text{, }&m=2\end{matrix}\right.$$