Simplify ((49x^2-25)/(2x^2+x-21))÷((7x+5)/(4x-12))

$\frac{49 x^{2} - 25}{2 x^{2} + x - 21} \div \frac{7 x + 5}{4 x - 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{49 x^{2} - 25}{2 x^{2} + x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Step 2

Simplify the numerator.

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$\frac{{(7 x)}^{2} - 25}{2 x^{2} + x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Rewrite $25$ as $5^{2}$ .

$\frac{{(7 x)}^{2} - 5^{2}}{2 x^{2} + x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7 x$ and $b = 5$ .

$\frac{(7 x + 5) (7 x - 5)}{2 x^{2} + x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Step 3

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{(7 x + 5) (7 x - 5)}{2 x^{2} + 1 x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Rewrite $1$ as $- 6$ plus $7$

$\frac{(7 x + 5) (7 x - 5)}{2 x^{2} + (- 6 + 7) x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Apply the distributive property.

$\frac{(7 x + 5) (7 x - 5)}{2 x^{2} - 6 x + 7 x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(7 x + 5) (7 x - 5)}{(2 x^{2} - 6 x) + 7 x - 21} \cdot \frac{4 x - 12}{7 x + 5}$

Factor out the greatest common factor (GCF) from each group.

$\frac{(7 x + 5) (7 x - 5)}{2 x (x - 3) + 7 (x - 3)} \cdot \frac{4 x - 12}{7 x + 5}$

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$\frac{(7 x + 5) (7 x - 5)}{(x - 3) (2 x + 7)} \cdot \frac{4 x - 12}{7 x + 5}$

Step 4

Cancel the common factor of $7 x + 5$ .

Cancel the common factor.

$\frac{(7 x + 5) (7 x - 5)}{(x - 3) (2 x + 7)} \cdot \frac{4 x - 12}{7 x + 5}$

Rewrite the expression.

$\frac{7 x - 5}{(x - 3) (2 x + 7)} (4 x - 12)$

Step 5

Multiply $\frac{7 x - 5}{(x - 3) (2 x + 7)}$ by $4 x - 12$ .

$\frac{(7 x - 5) (4 x - 12)}{(x - 3) (2 x + 7)}$

Step 6

Factor $4$ out of $4 x - 12$ .

Factor $4$ out of $4 x$ .

$\frac{(7 x - 5) (4 (x) - 12)}{(x - 3) (2 x + 7)}$

Factor $4$ out of $- 12$ .

$\frac{(7 x - 5) (4 x + 4 \cdot - 3)}{(x - 3) (2 x + 7)}$

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$\frac{(7 x - 5) (4 (x - 3))}{(x - 3) (2 x + 7)}$

$\frac{(7 x - 5) \cdot 4 (x - 3)}{(x - 3) (2 x + 7)}$

Step 7

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(7 x - 5) \cdot 4 (x - 3)}{(x - 3) (2 x + 7)}$

Rewrite the expression.

$\frac{(7 x - 5) \cdot 4}{2 x + 7}$

Step 8

Move $4$ to the left of $7 x - 5$ .

$\frac{4 (7 x - 5)}{2 x + 7}$

Simplify ((49x^2-25)/(2x^2+x-21))÷((7x+5)/(4x-12))

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