Algebra Examples

$\frac{3}{4 x^{2} + 4 x + 1} + \frac{x + 1}{4 x^{2} - 1}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{3}{{(2 x)}^{2} + 4 x + 1} + \frac{x + 1}{4 x^{2} - 1}$

Step 1.1.2

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

$\frac{3}{{(2 x)}^{2} + 4 x + 1^{2}} + \frac{x + 1}{4 x^{2} - 1}$

Step 1.1.3

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

$4 x = 2 \cdot (2 x) \cdot 1$

Step 1.1.4

Rewrite the polynomial.

$\frac{3}{{(2 x)}^{2} + 2 \cdot (2 x) \cdot 1 + 1^{2}} + \frac{x + 1}{4 x^{2} - 1}$

Step 1.1.5

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

$\frac{3}{{(2 x + 1)}^{2}} + \frac{x + 1}{4 x^{2} - 1}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 1$ .

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

Step 2

To write $\frac{3}{{(2 x + 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $3$ .

$\frac{6 x + 3 \cdot - 1 + (x + 1) (2 x + 1)}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.3

Multiply $3$ by $- 1$ .

$\frac{6 x - 3 + (x + 1) (2 x + 1)}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.4

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

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Step 6.4.1

Apply the distributive property.

$\frac{6 x - 3 + x (2 x + 1) + 1 (2 x + 1)}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.4.2

Apply the distributive property.

$\frac{6 x - 3 + x (2 x) + x \cdot 1 + 1 (2 x + 1)}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.4.3

Apply the distributive property.

$\frac{6 x - 3 + x (2 x) + x \cdot 1 + 1 (2 x) + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 x - 3 + 2 x \cdot x + x \cdot 1 + 1 (2 x) + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.1.2

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

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Step 6.5.1.2.1

Move $x$ .

$\frac{6 x - 3 + 2 (x \cdot x) + x \cdot 1 + 1 (2 x) + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.1.2.2

Multiply $x$ by $x$ .

$\frac{6 x - 3 + 2 x^{2} + x \cdot 1 + 1 (2 x) + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.1.3

Multiply $x$ by $1$ .

$\frac{6 x - 3 + 2 x^{2} + x + 1 (2 x) + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.1.4

Multiply $2 x$ by $1$ .

$\frac{6 x - 3 + 2 x^{2} + x + 2 x + 1 \cdot 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.1.5

Multiply $1$ by $1$ .

$\frac{6 x - 3 + 2 x^{2} + x + 2 x + 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.5.2

Add $x$ and $2 x$ .

$\frac{6 x - 3 + 2 x^{2} + 3 x + 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.6

Add $6 x$ and $3 x$ .

$\frac{9 x - 3 + 2 x^{2} + 1}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.7

Add $- 3$ and $1$ .

$\frac{9 x + 2 x^{2} - 2}{{(2 x + 1)}^{2} (2 x - 1)}$

Step 6.8

Reorder terms.

$\frac{2 x^{2} + 9 x - 2}{{(2 x + 1)}^{2} (2 x - 1)}$