Simplify (x^2-5x-36)/(x^2+10x+24)*(4x+24)/(6x-54)

$\frac{x^{2} - 5 x - 36}{x^{2} + 10 x + 24} \cdot \frac{4 x + 24}{6 x - 54}$

Step 1

Factor $x^{2} - 5 x - 36$ 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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Write the factored form using these integers.

$\frac{(x - 9) (x + 4)}{x^{2} + 10 x + 24} \cdot \frac{4 x + 24}{6 x - 54}$

Step 2

Factor $x^{2} + 10 x + 24$ 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 $10$ .

$4, 6$

Write the factored form using these integers.

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 x + 24}{6 x - 54}$

Step 3

Simplify terms.

Factor $4$ out of $4 x + 24$ .

Factor $4$ out of $4 x$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x) + 24}{6 x - 54}$

Factor $4$ out of $24$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 x + 4 \cdot 6}{6 x - 54}$

Factor $4$ out of $4 x + 4 \cdot 6$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{6 x - 54}$

Factor $6$ out of $6 x - 54$ .

Factor $6$ out of $6 x$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{6 (x) - 54}$

Factor $6$ out of $- 54$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{6 x + 6 \cdot - 9}$

Factor $6$ out of $6 x + 6 \cdot - 9$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{6 (x - 9)}$

Cancel the common factor of $x - 9$ .

Factor $x - 9$ out of $6 (x - 9)$ .

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{(x - 9) \cdot 6}$

Cancel the common factor.

$\frac{(x - 9) (x + 4)}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{(x - 9) \cdot 6}$

Rewrite the expression.

$\frac{x + 4}{(x + 4) (x + 6)} \cdot \frac{4 (x + 6)}{6}$

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $(x + 4) (x + 6)$ .

$\frac{x + 4}{(x + 6) (x + 4)} \cdot \frac{4 (x + 6)}{6}$

Factor $x + 6$ out of $4 (x + 6)$ .

$\frac{x + 4}{(x + 6) (x + 4)} \cdot \frac{(x + 6) \cdot 4}{6}$

Cancel the common factor.

$\frac{x + 4}{(x + 6) (x + 4)} \cdot \frac{(x + 6) \cdot 4}{6}$

Rewrite the expression.

$\frac{x + 4}{x + 4} \cdot \frac{4}{6}$

Multiply $\frac{x + 4}{x + 4}$ by $\frac{4}{6}$ .

$\frac{(x + 4) \cdot 4}{(x + 4) \cdot 6}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x + 4) \cdot 4}{(x + 4) \cdot 6}$

Rewrite the expression.

$\frac{4}{6}$

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$\frac{2 (2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 \cdot 2}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 3}$

Rewrite the expression.

$\frac{2}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Simplify (x^2-5x-36)/(x^2+10x+24)*(4x+24)/(6x-54)

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