Algebra Examples

$\frac{x}{2 - x} + \frac{3 x + 5}{x^{2} - 4}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x}{2 - x} + \frac{3 x + 5}{x^{2} - 2^{2}}$

Step 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 $b = 2$ .

$\frac{x}{2 - x} + \frac{3 x + 5}{(x + 2) (x - 2)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{x}{2 - x} + \frac{3 x + 5}{(x + 2) (- 1 (- x) - 2)}$

Step 2.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{x}{2 - x} + \frac{3 x + 5}{(x + 2) (- 1 (- x) - 1 (2))}$

Step 2.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

$\frac{x}{2 - x} + \frac{3 x + 5}{(x + 2) (- 1 (- x + 2))}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{3 x + 5}{(x + 2) (- 1 (- x + 2))}$ to the numerator.

$\frac{x}{2 - x} + \frac{- (3 x + 5)}{(x + 2) (- x + 2)}$

Step 2.4.2

Reorder terms.

$\frac{x}{- x + 2} + \frac{- (3 x + 5)}{(x + 2) (- x + 2)}$

Step 3

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

$\frac{x}{- x + 2} \cdot \frac{x + 2}{x + 2} + \frac{- (3 x + 5)}{(x + 2) (- x + 2)}$

Step 4

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

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Step 4.1

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

$\frac{x (x + 2)}{(- x + 2) (x + 2)} + \frac{- (3 x + 5)}{(x + 2) (- x + 2)}$

Step 4.2

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

$\frac{x (x + 2)}{(- x + 2) (x + 2)} + \frac{- (3 x + 5)}{(- x + 2) (x + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (x + 2) - (3 x + 5)}{(- x + 2) (x + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - (3 x + 5)}{(- x + 2) (x + 2)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - (3 x + 5)}{(- x + 2) (x + 2)}$

Step 6.3

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x - (3 x + 5)}{(- x + 2) (x + 2)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} + 2 x - (3 x) - 1 \cdot 5}{(- x + 2) (x + 2)}$

Step 6.5

Multiply $3$ by $- 1$ .

$\frac{x^{2} + 2 x - 3 x - 1 \cdot 5}{(- x + 2) (x + 2)}$

Step 6.6

Multiply $- 1$ by $5$ .

$\frac{x^{2} + 2 x - 3 x - 5}{(- x + 2) (x + 2)}$

Step 6.7

Subtract $3 x$ from $2 x$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- x$ .

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

Step 7.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{x^{2} - x - 5}{(- (x) - 1 (- 2)) (x + 2)}$

Step 7.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 7.4

Rewrite negatives.

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Step 7.4.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{x^{2} - x - 5}{- 1 (x - 2) (x + 2)}$

Step 7.4.2

Move the negative in front of the fraction.

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