Simplify ((7x^3)/(x^2+x-30))÷((x^2-6x)/(x^2-36))

$\frac{7 x^{3}}{x^{2} + x - 30} \div \frac{x^{2} - 6 x}{x^{2} - 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{7 x^{3}}{x^{2} + x - 30} \cdot \frac{x^{2} - 36}{x^{2} - 6 x}$

Step 2

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{x^{2} - 36}{x^{2} - 6 x}$

Step 3

Simplify the numerator.

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

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{x^{2} - 6^{2}}{x^{2} - 6 x}$

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 $b = 6$ .

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{(x + 6) (x - 6)}{x^{2} - 6 x}$

Step 4

Simplify terms.

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

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

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{(x + 6) (x - 6)}{x \cdot x - 6 x}$

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

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{(x + 6) (x - 6)}{x \cdot x + x \cdot - 6}$

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

$\frac{7 x^{3}}{(x - 5) (x + 6)} \cdot \frac{(x + 6) (x - 6)}{x (x - 6)}$

Combine.

$\frac{7 x^{3} ((x + 6) (x - 6))}{(x - 5) (x + 6) (x (x - 6))}$

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $7 x^{3} ((x + 6) (x - 6))$ .

$\frac{x (7 x^{2} ((x + 6) (x - 6)))}{(x - 5) (x + 6) (x (x - 6))}$

Cancel the common factors.

Factor $x$ out of $(x - 5) (x + 6) (x (x - 6))$ .

$\frac{x (7 x^{2} ((x + 6) (x - 6)))}{x ((x - 5) (x + 6) (x - 6))}$

Cancel the common factor.

$\frac{x (7 x^{2} ((x + 6) (x - 6)))}{x ((x - 5) (x + 6) (x - 6))}$

Rewrite the expression.

$\frac{7 x^{2} ((x + 6) (x - 6))}{(x - 5) (x + 6) (x - 6)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{7 x^{2} ((x + 6) (x - 6))}{(x - 5) (x + 6) (x - 6)}$

Rewrite the expression.

$\frac{7 x^{2} (x - 6)}{(x - 5) (x - 6)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{7 x^{2} (x - 6)}{(x - 5) (x - 6)}$

Rewrite the expression.

$\frac{7 x^{2}}{x - 5}$

Simplify ((7x^3)/(x^2+x-30))÷((x^2-6x)/(x^2-36))

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