Algebra Examples

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

Step 1

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

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Step 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 1.2

Factor $3$ out of $9$ .

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

Step 1.3

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 2

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

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Step 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 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 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 3

Cancel the common factor of $x + 3$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

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

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Step 4.1

Factor $4$ out of $8 x$ .

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

Step 4.2

Factor $4$ out of $12$ .

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

Step 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 5

Factor using the perfect square rule.

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Step 5.1

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

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

Step 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 5.3

Rewrite the polynomial.

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

Step 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 6

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2, {(x + 3)}^{2}$

Step 7

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 8

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 9

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 10

The LCM of $2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 11

The factors for $x + 3$ are $(x + 3) \cdot (x + 3)$ , which is $x + 3$ multiplied by itself $2$ times.

$(x + 3) = (x + 3) \cdot (x + 3)$

$(x + 3)$ occurs $2$ times.

Step 12

The LCM of ${(x + 3)}^{2}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(x + 3)}^{2}$

Step 13

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$2 {(x + 3)}^{2}$