Algebra Examples

$\frac{x + \frac{1}{x + 2}}{x - \frac{1}{x + 2}}$

Step 1

Multiply the numerator and denominator of the fraction by $x + 2$ .

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Step 1.1

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

$\frac{x + 2}{x + 2} \cdot \frac{x + \frac{1}{x + 2}}{x - \frac{1}{x + 2}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 2$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x + 2}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 4.3

Factor using the perfect square rule.

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Step 4.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 4.3.3

Rewrite the polynomial.

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

Step 4.3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $x$ by $x$ .

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