Algebra Examples

$\frac{3 x + 9}{2 x + 6} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x + 9$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 9}{2 x + 6} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.1.2

Factor $3$ out of $9$ .

$\frac{3 x + 3 \cdot 3}{2 x + 6} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.1.3

Factor $3$ out of $3 x + 3 \cdot 3$ .

$\frac{3 (x + 3)}{2 x + 6} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.2

Factor $2$ out of $2 x + 6$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

$\frac{3 (x + 3)}{2 (x) + 6} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.2.2

Factor $2$ out of $6$ .

$\frac{3 (x + 3)}{2 x + 2 \cdot 3} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.2.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{3 (x + 3)}{2 (x + 3)} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.3

Cancel the common factor of $x + 3$ .

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Step 1.3.1

Cancel the common factor.

$\frac{3 (x + 3)}{2 (x + 3)} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.3.2

Rewrite the expression.

$\frac{3}{2} + \frac{8 x + 12}{x^{2} + 6 x + 9}$

Step 1.4

Factor $4$ out of $8 x + 12$ .

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Step 1.4.1

Factor $4$ out of $8 x$ .

$\frac{3}{2} + \frac{4 (2 x) + 12}{x^{2} + 6 x + 9}$

Step 1.4.2

Factor $4$ out of $12$ .

$\frac{3}{2} + \frac{4 (2 x) + 4 (3)}{x^{2} + 6 x + 9}$

Step 1.4.3

Factor $4$ out of $4 (2 x) + 4 (3)$ .

$\frac{3}{2} + \frac{4 (2 x + 3)}{x^{2} + 6 x + 9}$

Step 1.5

Factor using the perfect square rule.

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Step 1.5.1

Rewrite $9$ as $3^{2}$ .

$\frac{3}{2} + \frac{4 (2 x + 3)}{x^{2} + 6 x + 3^{2}}$

Step 1.5.2

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

$6 x = 2 \cdot x \cdot 3$

Step 1.5.3

Rewrite the polynomial.

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

Step 1.5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 2

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

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

Step 3

To write $\frac{4 (2 x + 3)}{{(x + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $2 {(x + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $\frac{4 (2 x + 3)}{{(x + 3)}^{2}}$ by $\frac{2}{2}$ .

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

Step 4.3

Reorder the factors of ${(x + 3)}^{2} \cdot 2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

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

Step 6.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 6.3.1.2

Move $3$ to the left of $x$ .

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

Step 6.3.1.3

Multiply $3$ by $3$ .

$\frac{3 (x^{2} + 3 x + 3 x + 9) + 4 (2 x + 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.3.2

Add $3 x$ and $3 x$ .

$\frac{3 (x^{2} + 6 x + 9) + 4 (2 x + 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.4

Apply the distributive property.

$\frac{3 x^{2} + 3 (6 x) + 3 \cdot 9 + 4 (2 x + 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.5

Simplify.

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Step 6.5.1

Multiply $6$ by $3$ .

$\frac{3 x^{2} + 18 x + 3 \cdot 9 + 4 (2 x + 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.5.2

Multiply $3$ by $9$ .

$\frac{3 x^{2} + 18 x + 27 + 4 (2 x + 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.6

Apply the distributive property.

$\frac{3 x^{2} + 18 x + 27 + (4 (2 x) + 4 \cdot 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.7

Multiply $2$ by $4$ .

$\frac{3 x^{2} + 18 x + 27 + (8 x + 4 \cdot 3) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.8

Multiply $4$ by $3$ .

$\frac{3 x^{2} + 18 x + 27 + (8 x + 12) \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.9

Apply the distributive property.

$\frac{3 x^{2} + 18 x + 27 + 8 x \cdot 2 + 12 \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.10

Multiply $2$ by $8$ .

$\frac{3 x^{2} + 18 x + 27 + 16 x + 12 \cdot 2}{2 {(x + 3)}^{2}}$

Step 6.11

Multiply $12$ by $2$ .

$\frac{3 x^{2} + 18 x + 27 + 16 x + 24}{2 {(x + 3)}^{2}}$

Step 6.12

Add $18 x$ and $16 x$ .

$\frac{3 x^{2} + 34 x + 27 + 24}{2 {(x + 3)}^{2}}$

Step 6.13

Add $27$ and $24$ .

$\frac{3 x^{2} + 34 x + 51}{2 {(x + 3)}^{2}}$