Simplify ((x^2-64)/(x-8))÷((x^2+5x-24)/(3x))

$\frac{x^{2} - 64}{x - 8} \div \frac{x^{2} + 5 x - 24}{3 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 64}{x - 8} \cdot \frac{3 x}{x^{2} + 5 x - 24}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 8^{2}}{x - 8} \cdot \frac{3 x}{x^{2} + 5 x - 24}$

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 $b = 8$ .

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

Step 3

Cancel the common factor of $x - 8$ .

Cancel the common factor.

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

Divide $x + 8$ by $1$ .

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

Step 4

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.

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

Step 5

Cancel the common factor of $x + 8$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x^2-64)/(x-8))÷((x^2+5x-24)/(3x))

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