Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Add $\frac{3}{x + 2}$ to both sides of the equation.

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

Step 2

Simplify $\frac{2 x}{x^{2} - 4} - \frac{4}{x^{2} - 4} + \frac{3}{x + 2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the denominator.

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Step 2.1.1.1

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

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

Step 2.1.1.2

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 $b = 2$ .

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

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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 $b = 2$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify each term.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Multiply $3$ by $- 2$ .

$\frac{2 x + 3 x - 6 - 4}{(x + 2) (x - 2)} = 0$

Step 2.7

Add $2 x$ and $3 x$ .

$\frac{5 x - 6 - 4}{(x + 2) (x - 2)} = 0$

Step 2.8

Subtract $4$ from $- 6$ .

$\frac{5 x - 10}{(x + 2) (x - 2)} = 0$

Step 2.9

Factor $5$ out of $5 x - 10$ .

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Step 2.9.1

Factor $5$ out of $5 x$ .

$\frac{5 (x) - 10}{(x + 2) (x - 2)} = 0$

Step 2.9.2

Factor $5$ out of $- 10$ .

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

Step 2.9.3

Factor $5$ out of $5 x + 5 \cdot - 2$ .

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

Step 2.10

Cancel the common factor of $x - 2$ .

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Step 2.10.1

Cancel the common factor.

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

Step 2.10.2

Rewrite the expression.

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

Step 3

Set the numerator equal to zero.

$5 = 0$

Step 4

Since $5 \neq 0$ , there are no solutions.

No solution