Solve by Factoring 5/(x^2-4)+2/x=2/(x-2)

$\frac{5}{x^{2} - 4} + \frac{2}{x} = \frac{2}{x - 2}$

Step 1

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

$\frac{5}{x^{2} - 4} + \frac{2}{x} - \frac{2}{x - 2} = 0$

Step 2

Simplify $\frac{5}{x^{2} - 4} + \frac{2}{x} - \frac{2}{x - 2}$ .

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{5}{x^{2} - 2^{2}} + \frac{2}{x} - \frac{2}{x - 2} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{5}{(x + 2) (x - 2)} + \frac{2}{x} - \frac{2}{x - 2} = 0$

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

$\frac{2}{x} + \frac{5}{(x + 2) (x - 2)} - \frac{2}{x - 2} \cdot \frac{x + 2}{x + 2} = 0$

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

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

$\frac{2}{x} + \frac{5}{(x + 2) (x - 2)} - \frac{2 (x + 2)}{(x - 2) (x + 2)} = 0$

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

$\frac{2}{x} + \frac{5}{(x + 2) (x - 2)} - \frac{2 (x + 2)}{(x + 2) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{2}{x} + \frac{5 - 2 (x + 2)}{(x + 2) (x - 2)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2}{x} + \frac{5 - 2 x - 2 \cdot 2}{(x + 2) (x - 2)} = 0$

Multiply $- 2$ by $2$ .

$\frac{2}{x} + \frac{5 - 2 x - 4}{(x + 2) (x - 2)} = 0$

Subtract $4$ from $5$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $2$ .

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

Expand $(2 x + 4) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $4$ by $- 2$ .

$\frac{2 x^{2} - 4 x + 4 x - 8 + (- 2 x + 1) x}{x (x + 2) (x - 2)} = 0$

Add $- 4 x$ and $4 x$ .

$\frac{2 x^{2} + 0 - 8 + (- 2 x + 1) x}{x (x + 2) (x - 2)} = 0$

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

$\frac{2 x^{2} - 8 + (- 2 x + 1) x}{x (x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{2 x^{2} - 8 - 2 x \cdot x + 1 x}{x (x + 2) (x - 2)} = 0$

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

Move $x$ .

$\frac{2 x^{2} - 8 - 2 (x \cdot x) + 1 x}{x (x + 2) (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 8 - 2 x^{2} + 1 x}{x (x + 2) (x - 2)} = 0$

Multiply $x$ by $1$ .

$\frac{2 x^{2} - 8 - 2 x^{2} + x}{x (x + 2) (x - 2)} = 0$

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{0 - 8 + x}{x (x + 2) (x - 2)} = 0$

Subtract $8$ from $0$ .

$\frac{- 8 + x}{x (x + 2) (x - 2)} = 0$

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- 1 (8) + x}{x (x + 2) (x - 2)} = 0$

Factor $- 1$ out of $x$ .

$\frac{- 1 (8) - 1 (- x)}{x (x + 2) (x - 2)} = 0$

Factor $- 1$ out of $- 1 (8) - 1 (- x)$ .

$\frac{- 1 (8 - x)}{x (x + 2) (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{8 - x}{x (x + 2) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$8 - x = 0$

Step 4

Solve the equation for $x$ .

Subtract $8$ from both sides of the equation.

$- x = - 8$

Divide each term in $- x = - 8$ by $- 1$ and simplify.

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Simplify the right side.

Divide $- 8$ by $- 1$ .

$x = 8$

Solve by Factoring 5/(x^2-4)+2/x=2/(x-2)

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