Solve by Factoring (4x+1)/(x-2)+2/x=-4/(x^2-2x)

$\frac{4 x + 1}{x - 2} + \frac{2}{x} = \frac{- 4}{x^{2} - 2 x}$

Step 1

Subtract $\frac{- 4}{x^{2} - 2 x}$ from both sides of the equation.

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x} = 0$

Step 2

Simplify $\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x^{2} - 2 x}$ .

Simplify each term.

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

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

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x - 2 x} = 0$

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

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x \cdot x + x \cdot - 2} = 0$

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

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - \frac{- 4}{x (x - 2)} = 0$

Move the negative in front of the fraction.

$\frac{4 x + 1}{x - 2} + \frac{2}{x} - - \frac{4}{x (x - 2)} = 0$

Multiply $- - \frac{4}{x (x - 2)}$ .

Multiply $- 1$ by $- 1$ .

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + 1 \frac{4}{x (x - 2)} = 0$

Multiply $\frac{4}{x (x - 2)}$ by $1$ .

$\frac{4 x + 1}{x - 2} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

To write $\frac{4 x + 1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} + \frac{4}{x (x - 2)} = 0$

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

$\frac{4 x + 1}{x - 2} \cdot \frac{x}{x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0$

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

Multiply $\frac{4 x + 1}{x - 2}$ by $\frac{x}{x}$ .

$\frac{(4 x + 1) x}{(x - 2) x} + \frac{2}{x} \cdot \frac{x - 2}{x - 2} + \frac{4}{x (x - 2)} = 0$

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

$\frac{(4 x + 1) x}{(x - 2) x} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

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

$\frac{(4 x + 1) x}{x (x - 2)} + \frac{2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(4 x + 1) x + 2 (x - 2)}{x (x - 2)} + \frac{4}{x (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(4 x + 1) x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{4 x \cdot x + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

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

Move $x$ .

$\frac{4 (x \cdot x) + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 1 x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Multiply $x$ by $1$ .

$\frac{4 x^{2} + x + 2 (x - 2) + 4}{x (x - 2)} = 0$

Apply the distributive property.

$\frac{4 x^{2} + x + 2 x + 2 \cdot - 2 + 4}{x (x - 2)} = 0$

Multiply $2$ by $- 2$ .

$\frac{4 x^{2} + x + 2 x - 4 + 4}{x (x - 2)} = 0$

Combine the opposite terms in $4 x^{2} + x + 2 x - 4 + 4$ .

Add $- 4$ and $4$ .

$\frac{4 x^{2} + x + 2 x + 0}{x (x - 2)} = 0$

Add $4 x^{2} + x + 2 x$ and $0$ .

$\frac{4 x^{2} + x + 2 x}{x (x - 2)} = 0$

Add $x$ and $2 x$ .

$\frac{4 x^{2} + 3 x}{x (x - 2)} = 0$

Factor $x$ out of $4 x^{2} + 3 x$ .

Factor $x$ out of $4 x^{2}$ .

$\frac{x (4 x) + 3 x}{x (x - 2)} = 0$

Factor $x$ out of $3 x$ .

$\frac{x (4 x) + x \cdot 3}{x (x - 2)} = 0$

Factor $x$ out of $x (4 x) + x \cdot 3$ .

$\frac{x (4 x + 3)}{x (x - 2)} = 0$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (4 x + 3)}{x (x - 2)} = 0$

Rewrite the expression.

$\frac{4 x + 3}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$4 x + 3 = 0$

Step 4

Solve the equation for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Solve by Factoring (4x+1)/(x-2)+2/x=-4/(x^2-2x)

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