Algebra Examples

$\frac{3 x^{2} - 27}{x^{2} + 6 x + 9} + \frac{2 x - 4}{x^{2} + x - 6}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Factor $3$ out of $3 x^{2} - 27$ .

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Step 1.1.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 27}{x^{2} + 6 x + 9} + \frac{2 x - 4}{x^{2} + x - 6}$

Step 1.1.1.2

Factor $3$ out of $- 27$ .

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

Step 1.1.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 9$ .

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

Step 1.1.2

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

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

Step 1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

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

Step 1.2.2

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 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

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

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

Step 1.3

Cancel the common factor of $x + 3$ and ${(x + 3)}^{2}$ .

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Step 1.3.1

Factor $x + 3$ out of $3 (x + 3) (x - 3)$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $x + 3$ out of ${(x + 3)}^{2}$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

Factor $2$ out of $2 x - 4$ .

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Step 1.4.1

Factor $2$ out of $2 x$ .

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

Step 1.4.2

Factor $2$ out of $- 4$ .

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

Step 1.4.3

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

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

Step 1.5

Factor $x^{2} + x - 6$ using the AC method.

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Step 1.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.5.2

Write the factored form using these integers.

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

Step 1.6

Cancel the common factor of $x - 2$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 3 + 2}{x + 3}$

Step 3.2

Multiply $3$ by $- 3$ .

$\frac{3 x - 9 + 2}{x + 3}$

Step 4

Add $- 9$ and $2$ .

$\frac{3 x - 7}{x + 3}$