Simplify (x^3+9x^2)/(2x)*(x^2-14x+45)/(x^2-81)

$\frac{x^{3} + 9 x^{2}}{2 x} \cdot \frac{x^{2} - 14 x + 45}{x^{2} - 81}$

Step 1

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

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

$\frac{x^{2} x + 9 x^{2}}{2 x} \cdot \frac{x^{2} - 14 x + 45}{x^{2} - 81}$

Factor $x^{2}$ out of $9 x^{2}$ .

$\frac{x^{2} x + x^{2} \cdot 9}{2 x} \cdot \frac{x^{2} - 14 x + 45}{x^{2} - 81}$

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

$\frac{x^{2} (x + 9)}{2 x} \cdot \frac{x^{2} - 14 x + 45}{x^{2} - 81}$

Step 2

Factor $x^{2} - 14 x + 45$ 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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Write the factored form using these integers.

$\frac{x^{2} (x + 9)}{2 x} \cdot \frac{(x - 9) (x - 5)}{x^{2} - 81}$

Step 3

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

$\frac{x^{2} (x + 9)}{2 x} \cdot \frac{(x - 9) (x - 5)}{x^{2} - 9^{2}}$

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 $b = 9$ .

$\frac{x^{2} (x + 9)}{2 x} \cdot \frac{(x - 9) (x - 5)}{(x + 9) (x - 9)}$

Step 4

Simplify terms.

Combine.

$\frac{x^{2} (x + 9) ((x - 9) (x - 5))}{2 x ((x + 9) (x - 9))}$

Cancel the common factor of $x^{2}$ and $x$ .

Factor $x$ out of $x^{2} (x + 9) ((x - 9) (x - 5))$ .

$\frac{x ((x (x + 9)) ((x - 9) (x - 5)))}{2 x ((x + 9) (x - 9))}$

Cancel the common factors.

Factor $x$ out of $2 x ((x + 9) (x - 9))$ .

$\frac{x ((x (x + 9)) ((x - 9) (x - 5)))}{x (2 ((x + 9) (x - 9)))}$

Cancel the common factor.

$\frac{x ((x (x + 9)) ((x - 9) (x - 5)))}{x (2 ((x + 9) (x - 9)))}$

Rewrite the expression.

$\frac{(x (x + 9)) ((x - 9) (x - 5))}{2 ((x + 9) (x - 9))}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{x (x + 9) ((x - 9) (x - 5))}{2 ((x + 9) (x - 9))}$

Rewrite the expression.

$\frac{(x) ((x - 9) (x - 5))}{2 (x - 9)}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{x ((x - 9) (x - 5))}{2 (x - 9)}$

Rewrite the expression.

$\frac{(x) (x - 5)}{2}$

$\frac{x (x - 5)}{2}$

Simplify (x^3+9x^2)/(2x)*(x^2-14x+45)/(x^2-81)

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