Simplify (x^2+7x+12)/(x-3)*(x^2-6x+9)/(2x^2-18)

$\frac{x^{2} + 7 x + 12}{x - 3} \cdot \frac{x^{2} - 6 x + 9}{2 x^{2} - 18}$

Step 1

Factor $x^{2} + 7 x + 12$ using the AC method.

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 $12$ and whose sum is $7$ .

$3, 4$

Write the factored form using these integers.

$\frac{(x + 3) (x + 4)}{x - 3} \cdot \frac{x^{2} - 6 x + 9}{2 x^{2} - 18}$

Step 2

Factor using the perfect square rule.

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

$\frac{(x + 3) (x + 4)}{x - 3} \cdot \frac{x^{2} - 6 x + 3^{2}}{2 x^{2} - 18}$

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

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

Rewrite the polynomial.

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

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

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

Step 3

Simplify the denominator.

Factor $2$ out of $2 x^{2} - 18$ .

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

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

Factor $2$ out of $- 18$ .

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

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

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

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

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

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 $b = 3$ .

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

Step 4

Simplify terms.

Cancel the common factor of $x + 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Combine $x$ and $\frac{1}{2}$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

$\frac{x}{2} + 2 \cdot 2 \frac{1}{2}$

Rewrite the expression.

$\frac{x}{2} + 2$

Step 5

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6

Combine $2$ and $\frac{2}{2}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Multiply $2$ by $2$ .

$\frac{x + 4}{2}$

Simplify (x^2+7x+12)/(x-3)*(x^2-6x+9)/(2x^2-18)

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