Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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 $b = 2$ .

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

Step 2

To write $\frac{3}{5 x + 2}$ as a fraction with a common denominator, multiply by $\frac{5 x - 2}{5 x - 2}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{5 x + 2}$ by $\frac{5 x - 2}{5 x - 2}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $5$ by $3$ .

$\frac{15 x + 3 \cdot - 2 + 5 x}{(5 x + 2) (5 x - 2)}$

Step 4.3

Multiply $3$ by $- 2$ .

$\frac{15 x - 6 + 5 x}{(5 x + 2) (5 x - 2)}$

Step 4.4

Add $15 x$ and $5 x$ .

$\frac{20 x - 6}{(5 x + 2) (5 x - 2)}$

Step 4.5

Factor $2$ out of $20 x - 6$ .

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Step 4.5.1

Factor $2$ out of $20 x$ .

$\frac{2 (10 x) - 6}{(5 x + 2) (5 x - 2)}$

Step 4.5.2

Factor $2$ out of $- 6$ .

$\frac{2 (10 x) + 2 (- 3)}{(5 x + 2) (5 x - 2)}$

Step 4.5.3

Factor $2$ out of $2 (10 x) + 2 (- 3)$ .

$\frac{2 (10 x - 3)}{(5 x + 2) (5 x - 2)}$