Divide ((25x^2-4)/(x^2-9))÷((5x-2)/(x+3))

$\frac{25 x^{2} - 4}{x^{2} - 9} \div \frac{5 x - 2}{x + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{25 x^{2} - 4}{x^{2} - 9} \cdot \frac{x + 3}{5 x - 2}$

Step 2

Simplify the numerator.

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{{(5 x)}^{2} - 4}{x^{2} - 9} \cdot \frac{x + 3}{5 x - 2}$

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

$\frac{{(5 x)}^{2} - 2^{2}}{x^{2} - 9} \cdot \frac{x + 3}{5 x - 2}$

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

$\frac{(5 x + 2) (5 x - 2)}{x^{2} - 9} \cdot \frac{x + 3}{5 x - 2}$

Step 3

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

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

Step 4

Cancel the common factor of $5 x - 2$ .

Factor $5 x - 2$ out of $(5 x + 2) (5 x - 2)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{5 x + 2}{x - 3}$