Solve 2x/x-3-x^2/x-2+3x-x^2/2-x=x-x^2/x^2-5x+6

$$\frac { 2 x } { x - 3 } - \frac { x ^ { 2 } } { x - 2 } + \frac { 3 x - x ^ { 2 } } { 2 - x } = \frac { x - x ^ { 2 } } { x ^ { 2 } - 5 x + 6 }$$

Solve for x

$$x=0$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right), the least common multiple of x-3,x-2,2-x,x^{2}-5x+6.

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

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

$$\left(2x-4\right)x-\left(x-3\right)x^{2}+\left(3-x\right)\left(3x-x^{2}\right)=x-x^{2}$$

Use the distributive property to multiply 2x-4 by x.

$$2x^{2}-4x-\left(x-3\right)x^{2}+\left(3-x\right)\left(3x-x^{2}\right)=x-x^{2}$$

Use the distributive property to multiply x-3 by x^{2}.

$$2x^{2}-4x-\left(x^{3}-3x^{2}\right)+\left(3-x\right)\left(3x-x^{2}\right)=x-x^{2}$$

To find the opposite of x^{3}-3x^{2}, find the opposite of each term.

$$2x^{2}-4x-x^{3}+3x^{2}+\left(3-x\right)\left(3x-x^{2}\right)=x-x^{2}$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}-4x-x^{3}+\left(3-x\right)\left(3x-x^{2}\right)=x-x^{2}$$

Use the distributive property to multiply 3-x by 3x-x^{2} and combine like terms.

$$5x^{2}-4x-x^{3}+9x-6x^{2}+x^{3}=x-x^{2}$$

Combine -4x and 9x to get 5x.

$$5x^{2}+5x-x^{3}-6x^{2}+x^{3}=x-x^{2}$$

Combine 5x^{2} and -6x^{2} to get -x^{2}.

$$-x^{2}+5x-x^{3}+x^{3}=x-x^{2}$$

Combine -x^{3} and x^{3} to get 0.

$$-x^{2}+5x=x-x^{2}$$

Subtract x from both sides.

$$-x^{2}+5x-x=-x^{2}$$

Combine 5x and -x to get 4x.

$$-x^{2}+4x=-x^{2}$$

Add x^{2} to both sides.

$$-x^{2}+4x+x^{2}=0$$

Combine -x^{2} and x^{2} to get 0.

$$4x=0$$

Product of two numbers is equal to 0 if at least one of them is 0. Since 4 is not equal to 0, x must be equal to 0.

$$x=0$$