Algebra Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

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

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

Step 1.2

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

$4 x = 2 \cdot x \cdot 2$

Step 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Raise $x + 2$ to the power of $1$ .

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

Step 4.4

Raise $x + 2$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Reorder the factors of ${(x + 2)}^{2} x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Move $x$ .

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

Step 6.1.2

Multiply $x$ by $x$ .

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

Step 6.2

Expand $(x - 1) (x + 2)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $x$ by $x$ .

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

Step 6.3.1.2

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

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

Step 6.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.3.1.4

Multiply $- 1$ by $2$ .

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

Step 6.3.2

Subtract $x$ from $2 x$ .

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

Step 6.4

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

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

Step 6.5

Factor by grouping.

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Step 6.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Step 6.5.1.1

Multiply by $1$ .

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

Step 6.5.1.2

Rewrite $1$ as $- 2$ plus $3$

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

Step 6.5.1.3

Apply the distributive property.

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

Step 6.5.2

Factor out the greatest common factor from each group.

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Step 6.5.2.1

Group the first two terms and the last two terms.

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

Step 6.5.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 6.5.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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