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

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{1}{x^{2} - 4}$ from both sides of the equation.

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

Subtract $1$ from both sides of the equation.

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

Step 2

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

Simplify the denominator.

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

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

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 $b = 2$ .

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

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

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

Combine $- 1$ and $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $2 x$ .

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

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

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

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

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

Add $- 1$ and $4$ .

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

Factor $x^{2} + 4 x + 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Step 3

Set the numerator equal to zero.

$(x + 1) (x + 3) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x + 3 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$

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

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