Simplify (2x^2-72)/(6x^2+12x-288)*(x^2+5x-24)/(x^2+3x-18)

$\frac{2 x^{2} - 72}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Step 1

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 72$ .

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

$\frac{2 (x^{2}) - 72}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $2$ out of $- 72$ .

$\frac{2 x^{2} + 2 \cdot - 36}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $2$ out of $2 x^{2} + 2 \cdot - 36$ .

$\frac{2 (x^{2} - 36)}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Rewrite $36$ as $6^{2}$ .

$\frac{2 (x^{2} - 6^{2})}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

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 $b = 6$ .

$\frac{2 (x + 6) (x - 6)}{6 x^{2} + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Step 2

Simplify the denominator.

Factor $6$ out of $6 x^{2} + 12 x - 288$ .

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

$\frac{2 (x + 6) (x - 6)}{6 (x^{2}) + 12 x - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $6$ out of $12 x$ .

$\frac{2 (x + 6) (x - 6)}{6 (x^{2}) + 6 (2 x) - 288} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $6$ out of $- 288$ .

$\frac{2 (x + 6) (x - 6)}{6 x^{2} + 6 (2 x) + 6 \cdot - 48} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

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

$\frac{2 (x + 6) (x - 6)}{6 (x^{2} + 2 x) + 6 \cdot - 48} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $6$ out of $6 (x^{2} + 2 x) + 6 \cdot - 48$ .

$\frac{2 (x + 6) (x - 6)}{6 (x^{2} + 2 x - 48)} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Factor $x^{2} + 2 x - 48$ 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 $- 48$ and whose sum is $2$ .

$- 6, 8$

Write the factored form using these integers.

$\frac{2 (x + 6) (x - 6)}{6 ((x - 6) (x + 8))} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

$\frac{2 (x + 6) (x - 6)}{6 (x - 6) (x + 8)} \cdot \frac{x^{2} + 5 x - 24}{x^{2} + 3 x - 18}$

Step 3

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$\frac{2 (x + 6) (x - 6)}{6 (x - 6) (x + 8)} \cdot \frac{(x - 3) (x + 8)}{x^{2} + 3 x - 18}$

Step 4

Factor $x^{2} + 3 x - 18$ 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

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

Step 5

Simplify terms.

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $2 (x + 6) (x - 6)$ .

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

Factor $x + 6$ out of $(x - 3) (x + 6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $6 (x - 6) (x + 8)$ .

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

Factor $x + 8$ out of $(x - 3) (x + 8)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{2 (x - 6)}{6 (x - 6)}$ by $\frac{x - 3}{x - 3}$ .

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

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

Factor $2$ out of $2 (x - 6) (x - 3)$ .

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

Cancel the common factors.

Factor $2$ out of $6 (x - 6) (x - 3)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x - 6) (x - 3)}{3 (x - 6) (x - 3)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{(x - 6) (x - 3)}{3 (x - 6) (x - 3)}$

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Simplify (2x^2-72)/(6x^2+12x-288)*(x^2+5x-24)/(x^2+3x-18)

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