Simplify (2x^2-18x)/(6x^4-102x^3+432x^2)*(x^2-64)/(x^2+3x-40)

$\frac{2 x^{2} - 18 x}{6 x^{4} - 102 x^{3} + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Step 1

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

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

$\frac{2 x (x) - 18 x}{6 x^{4} - 102 x^{3} + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $2 x$ out of $- 18 x$ .

$\frac{2 x (x) + 2 x (- 9)}{6 x^{4} - 102 x^{3} + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

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

$\frac{2 x (x - 9)}{6 x^{4} - 102 x^{3} + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Step 2

Simplify the denominator.

Factor $6 x^{2}$ out of $6 x^{4} - 102 x^{3} + 432 x^{2}$ .

Factor $6 x^{2}$ out of $6 x^{4}$ .

$\frac{2 x (x - 9)}{6 x^{2} (x^{2}) - 102 x^{3} + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $6 x^{2}$ out of $- 102 x^{3}$ .

$\frac{2 x (x - 9)}{6 x^{2} (x^{2}) + 6 x^{2} (- 17 x) + 432 x^{2}} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $6 x^{2}$ out of $432 x^{2}$ .

$\frac{2 x (x - 9)}{6 x^{2} (x^{2}) + 6 x^{2} (- 17 x) + 6 x^{2} (72)} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2}) + 6 x^{2} (- 17 x)$ .

$\frac{2 x (x - 9)}{6 x^{2} (x^{2} - 17 x) + 6 x^{2} (72)} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2} - 17 x) + 6 x^{2} (72)$ .

$\frac{2 x (x - 9)}{6 x^{2} (x^{2} - 17 x + 72)} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Factor $x^{2} - 17 x + 72$ 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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Write the factored form using these integers.

$\frac{2 x (x - 9)}{6 x^{2} ((x - 9) (x - 8))} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

$\frac{2 x (x - 9)}{6 x^{2} (x - 9) (x - 8)} \cdot \frac{x^{2} - 64}{x^{2} + 3 x - 40}$

Step 3

Simplify the numerator.

Rewrite $64$ as $8^{2}$ .

$\frac{2 x (x - 9)}{6 x^{2} (x - 9) (x - 8)} \cdot \frac{x^{2} - 8^{2}}{x^{2} + 3 x - 40}$

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 $b = 8$ .

$\frac{2 x (x - 9)}{6 x^{2} (x - 9) (x - 8)} \cdot \frac{(x + 8) (x - 8)}{x^{2} + 3 x - 40}$

Step 4

Factor $x^{2} + 3 x - 40$ 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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Write the factored form using these integers.

$\frac{2 x (x - 9)}{6 x^{2} (x - 9) (x - 8)} \cdot \frac{(x + 8) (x - 8)}{(x - 5) (x + 8)}$

Step 5

Simplify terms.

Combine.

$\frac{2 x (x - 9) ((x + 8) (x - 8))}{6 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 x (x - 9) ((x + 8) (x - 8))$ .

$\frac{2 ((x (x - 9)) ((x + 8) (x - 8)))}{6 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))}$

Cancel the common factors.

Factor $2$ out of $6 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))$ .

$\frac{2 ((x (x - 9)) ((x + 8) (x - 8)))}{2 (3 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8)))}$

Cancel the common factor.

$\frac{2 ((x (x - 9)) ((x + 8) (x - 8)))}{2 (3 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8)))}$

Rewrite the expression.

$\frac{(x (x - 9)) ((x + 8) (x - 8))}{3 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))}$

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

Factor $x$ out of $(x (x - 9)) ((x + 8) (x - 8))$ .

$\frac{x ((x - 9) ((x + 8) (x - 8)))}{3 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))}$

Cancel the common factors.

Factor $x$ out of $3 x^{2} (x - 9) (x - 8) ((x - 5) (x + 8))$ .

$\frac{x ((x - 9) ((x + 8) (x - 8)))}{x (3 x (x - 9) (x - 8) ((x - 5) (x + 8)))}$

Cancel the common factor.

$\frac{x ((x - 9) ((x + 8) (x - 8)))}{x (3 x (x - 9) (x - 8) ((x - 5) (x + 8)))}$

Rewrite the expression.

$\frac{(x - 9) ((x + 8) (x - 8))}{3 x (x - 9) (x - 8) ((x - 5) (x + 8))}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{(x - 9) ((x + 8) (x - 8))}{3 x (x - 9) (x - 8) ((x - 5) (x + 8))}$

Rewrite the expression.

$\frac{(x + 8) (x - 8)}{(3 x (x - 8)) ((x - 5) (x + 8))}$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{(x + 8) (x - 8)}{3 x (x - 8) ((x - 5) (x + 8))}$

Rewrite the expression.

$\frac{x - 8}{(3 x (x - 8)) (x - 5)}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{x - 8}{3 x (x - 8) (x - 5)}$

Rewrite the expression.

$\frac{1}{(3 x) (x - 5)}$

$\frac{1}{3 x (x - 5)}$

Simplify (2x^2-18x)/(6x^4-102x^3+432x^2)*(x^2-64)/(x^2+3x-40)

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