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

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

Solve for a

$$a=\frac{2x^{2}+6x+3}{x^{2}+3x+2}$$

$$x\neq -2\text{ and }x\neq -1$$

Steps for Solving Linear Equation

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

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

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

$$ax^{2}-2x^{2}+3ax-6x+2a-3=0$$

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

$$ax^{2}+3ax-6x+2a-3=2x^{2}$$

Add 6x to both sides.

$$ax^{2}+3ax+2a-3=2x^{2}+6x$$

Add 3 to both sides.

$$ax^{2}+3ax+2a=2x^{2}+6x+3$$

Combine all terms containing a.

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

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

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

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

$$a=\frac{2x^{2}+6x+3}{x^{2}+3x+2}$$

Solve for x (complex solution)

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

$$x=\frac{-\sqrt{\left(a-6\right)\left(a-2\right)}-3a+6}{2\left(a-2\right)}\text{, }a\neq 2$$

Solve for x

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

$$x=\frac{-\sqrt{\left(a-6\right)\left(a-2\right)}-3a+6}{2\left(a-2\right)}\text{, }a\geq 6\text{ or }a<2$$