Divide ((4x)/(x^2+x-42))÷((x^3)/(x^2-49))

$\frac{4 x}{x^{2} + x - 42} \div \frac{x^{3}}{x^{2} - 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{3}}$

Step 2

Cancel the common factor of $x$ .

Factor $x$ out of $4 x$ .

$\frac{x \cdot 4}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{3}}$

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot 4}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x \cdot x^{2}}$

Cancel the common factor.

$\frac{x \cdot 4}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x \cdot x^{2}}$

Rewrite the expression.

$\frac{4}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2}}$

Step 3

Multiply $\frac{4}{x^{2} + x - 42}$ by $\frac{x^{2} - 49}{x^{2}}$ .

$\frac{4 (x^{2} - 49)}{(x^{2} + x - 42) x^{2}}$

Step 4

Simplify the numerator.

Rewrite $49$ as $7^{2}$ .

$\frac{4 (x^{2} - 7^{2})}{(x^{2} + x - 42) x^{2}}$

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 $b = 7$ .

$\frac{4 (x + 7) (x - 7)}{(x^{2} + x - 42) x^{2}}$

Step 5

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Write the factored form using these integers.

$\frac{4 (x + 7) (x - 7)}{(x - 6) (x + 7) x^{2}}$

Step 6

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{4 (x + 7) (x - 7)}{(x - 6) (x + 7) x^{2}}$

Rewrite the expression.

$\frac{4 (x - 7)}{(x - 6) x^{2}}$

Step 7

Reorder factors in $\frac{4 (x - 7)}{(x - 6) x^{2}}$ .

$\frac{4 (x - 7)}{x^{2} (x - 6)}$

Step 8

Apply the distributive property.

$\frac{4 x + 4 \cdot - 7}{x^{2} (x - 6)}$

Step 9

Multiply $4$ by $- 7$ .

$\frac{4 x - 28}{x^{2} (x - 6)}$

Step 10

Split the fraction $\frac{4 x - 28}{x^{2} (x - 6)}$ into two fractions.

$\frac{4 x}{x^{2} (x - 6)} + \frac{- 28}{x^{2} (x - 6)}$

Step 11

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

Factor $x$ out of $4 x$ .

$\frac{x \cdot 4}{x^{2} (x - 6)} + \frac{- 28}{x^{2} (x - 6)}$

Cancel the common factors.

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

$\frac{x \cdot 4}{x (x (x - 6))} + \frac{- 28}{x^{2} (x - 6)}$

Cancel the common factor.

$\frac{x \cdot 4}{x (x (x - 6))} + \frac{- 28}{x^{2} (x - 6)}$

Rewrite the expression.

$\frac{4}{x (x - 6)} + \frac{- 28}{x^{2} (x - 6)}$

Step 12

Move the negative in front of the fraction.

$\frac{4}{x (x - 6)} - \frac{28}{x^{2} (x - 6)}$

Divide ((4x)/(x^2+x-42))÷((x^3)/(x^2-49))

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