Algebra Examples

$\frac{4 x + 16}{25 x^{2} - 9} \div \frac{3 x + 2}{5 x - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x + 16}{25 x^{2} - 9} \cdot \frac{5 x - 3}{3 x + 2}$

Step 2

Factor $4$ out of $4 x + 16$ .

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Step 2.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) + 16}{25 x^{2} - 9} \cdot \frac{5 x - 3}{3 x + 2}$

Step 2.2

Factor $4$ out of $16$ .

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

Step 2.3

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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 $b = 3$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $5 x - 3$ .

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Step 4.1.1

Factor $5 x - 3$ out of $(5 x + 3) (5 x - 3)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

$\frac{4 (x + 4)}{5 x + 3} \cdot \frac{1}{3 x + 2}$

Step 4.2

Multiply $\frac{4 (x + 4)}{5 x + 3}$ by $\frac{1}{3 x + 2}$ .

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