Solve by Factoring (3x+2)/(x-2)+1/x=-2/(x^2-2x)

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

Step 1

Subtract $\frac{- 2}{x^{2} - 2 x}$ from both sides of the equation.

$\frac{3 x + 2}{x - 2} + \frac{1}{x} - \frac{- 2}{x^{2} - 2 x} = 0$

Step 2

Simplify $\frac{3 x + 2}{x - 2} + \frac{1}{x} - \frac{- 2}{x^{2} - 2 x}$ .

Simplify each term.

Factor $x$ out of $x^{2} - 2 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{3 x + 2}{x - 2} + \frac{1}{x} - \frac{- 2}{x \cdot x - 2 x} = 0$

Factor $x$ out of $- 2 x$ .

$\frac{3 x + 2}{x - 2} + \frac{1}{x} - \frac{- 2}{x \cdot x + x \cdot - 2} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$\frac{3 x + 2}{x - 2} + \frac{1}{x} - \frac{- 2}{x (x - 2)} = 0$

Move the negative in front of the fraction.

$\frac{3 x + 2}{x - 2} + \frac{1}{x} - - \frac{2}{x (x - 2)} = 0$

Multiply $- - \frac{2}{x (x - 2)}$ .

Multiply $- 1$ by $- 1$ .

$\frac{3 x + 2}{x - 2} + \frac{1}{x} + 1 \frac{2}{x (x - 2)} = 0$

Multiply $\frac{2}{x (x - 2)}$ by $1$ .

$\frac{3 x + 2}{x - 2} + \frac{1}{x} + \frac{2}{x (x - 2)} = 0$

To write $\frac{3 x + 2}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 x + 2}{x - 2} \cdot \frac{x}{x} + \frac{1}{x} + \frac{2}{x (x - 2)} = 0$

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{3 x + 2}{x - 2} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{x - 2}{x - 2} + \frac{2}{x (x - 2)} = 0$

Write each expression with a common denominator of $(x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x + 2}{x - 2}$ by $\frac{x}{x}$ .

$\frac{(3 x + 2) x}{(x - 2) x} + \frac{1}{x} \cdot \frac{x - 2}{x - 2} + \frac{2}{x (x - 2)} = 0$

Multiply $\frac{1}{x}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(3 x + 2) x}{(x - 2) x} + \frac{x - 2}{x (x - 2)} + \frac{2}{x (x - 2)} = 0$

Reorder the factors of $(x - 2) x$ .

$\frac{(3 x + 2) x}{x (x - 2)} + \frac{x - 2}{x (x - 2)} + \frac{2}{x (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(3 x + 2) x + x - 2}{x (x - 2)} + \frac{2}{x (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(3 x + 2) x + x - 2 + 2}{x (x - 2)} = 0$

Combine the opposite terms in $(3 x + 2) x + x - 2 + 2$ .

Add $- 2$ and $2$ .

$\frac{(3 x + 2) x + x + 0}{x (x - 2)} = 0$

Add $(3 x + 2) x + x$ and $0$ .

$\frac{(3 x + 2) x + x}{x (x - 2)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{3 x \cdot x + 2 x + x}{x (x - 2)} = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 (x \cdot x) + 2 x + x}{x (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 2 x + x}{x (x - 2)} = 0$

Add $2 x$ and $x$ .

$\frac{3 x^{2} + 3 x}{x (x - 2)} = 0$

Factor $3 x$ out of $3 x^{2} + 3 x$ .

Factor $3 x$ out of $3 x^{2}$ .

$\frac{3 x (x) + 3 x}{x (x - 2)} = 0$

Factor $3 x$ out of $3 x$ .

$\frac{3 x (x) + 3 x (1)}{x (x - 2)} = 0$

Factor $3 x$ out of $3 x (x) + 3 x (1)$ .

$\frac{3 x (x + 1)}{x (x - 2)} = 0$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{3 x (x + 1)}{x (x - 2)} = 0$

Rewrite the expression.

$\frac{3 (x + 1)}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$3 (x + 1) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (x + 1) = 0$ by $3$ and simplify.

Divide each term in $3 (x + 1) = 0$ by $3$ .

$\frac{3 (x + 1)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x + 1)}{3} = \frac{0}{3}$

Divide $x + 1$ by $1$ .

$x + 1 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Solve by Factoring (3x+2)/(x-2)+1/x=-2/(x^2-2x)

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