Simplify ((9x^2-4)/(5x^2-3x-14))÷((3x-2)/(3x-6))

$\frac{9 x^{2} - 4}{5 x^{2} - 3 x - 14} \div \frac{3 x - 2}{3 x - 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x^{2} - 4}{5 x^{2} - 3 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Step 2

Simplify the numerator.

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{{(3 x)}^{2} - 4}{5 x^{2} - 3 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Rewrite $4$ as $2^{2}$ .

$\frac{{(3 x)}^{2} - 2^{2}}{5 x^{2} - 3 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 2$ .

$\frac{(3 x + 2) (3 x - 2)}{5 x^{2} - 3 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

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 $a \cdot c = 5 \cdot - 14 = - 70$ and whose sum is $b = - 3$ .

Factor $- 3$ out of $- 3 x$ .

$\frac{(3 x + 2) (3 x - 2)}{5 x^{2} - 3 (x) - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Rewrite $- 3$ as $7$ plus $- 10$

$\frac{(3 x + 2) (3 x - 2)}{5 x^{2} + (7 - 10) x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Apply the distributive property.

$\frac{(3 x + 2) (3 x - 2)}{5 x^{2} + 7 x - 10 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x + 2) (3 x - 2)}{(5 x^{2} + 7 x) - 10 x - 14} \cdot \frac{3 x - 6}{3 x - 2}$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 x + 7$ .

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

Step 4

Cancel the common factor of $3 x - 2$ .

Factor $3 x - 2$ out of $(3 x + 2) (3 x - 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Multiply $\frac{3 x + 2}{(5 x + 7) (x - 2)}$ by $3 x - 6$ .

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

Step 6

Factor $3$ out of $3 x - 6$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 6$ .

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

Factor $3$ out of $3 x + 3 \cdot - 2$ .

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

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

Step 7

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Move $3$ to the left of $3 x + 2$ .

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

Simplify ((9x^2-4)/(5x^2-3x-14))÷((3x-2)/(3x-6))

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