Precalculus Examples

$\frac{7 a}{6 a^{2} - 15 a} + \frac{12 a}{4 a^{2} - 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $3 a$ out of $6 a^{2} - 15 a$ .

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Step 1.1.1

Factor $3 a$ out of $6 a^{2}$ .

$\frac{7 a}{3 a (2 a) - 15 a} + \frac{12 a}{4 a^{2} - 25}$

Step 1.1.2

Factor $3 a$ out of $- 15 a$ .

$\frac{7 a}{3 a (2 a) + 3 a (- 5)} + \frac{12 a}{4 a^{2} - 25}$

Step 1.1.3

Factor $3 a$ out of $3 a (2 a) + 3 a (- 5)$ .

$\frac{7 a}{3 a (2 a - 5)} + \frac{12 a}{4 a^{2} - 25}$

Step 1.2

Cancel the common factor of $a$ .

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Step 1.2.1

Cancel the common factor.

$\frac{7 a}{3 a (2 a - 5)} + \frac{12 a}{4 a^{2} - 25}$

Step 1.2.2

Rewrite the expression.

$\frac{7}{3 (2 a - 5)} + \frac{12 a}{4 a^{2} - 25}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$\frac{7}{3 (2 a - 5)} + \frac{12 a}{{(2 a)}^{2} - 25}$

Step 1.3.2

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

$\frac{7}{3 (2 a - 5)} + \frac{12 a}{{(2 a)}^{2} - 5^{2}}$

Step 1.3.3

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

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

Step 2

To write $\frac{7}{3 (2 a - 5)}$ as a fraction with a common denominator, multiply by $\frac{2 a + 5}{2 a + 5}$ .

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

Step 3

To write $\frac{12 a}{(2 a + 5) (2 a - 5)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $3 (2 a - 5) (2 a + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $3 (2 a - 5) (2 a + 5)$ .

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

Step 4.4

Reorder the factors of $(2 a + 5) (2 a - 5) \cdot 3$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $7$ .

$\frac{14 a + 7 \cdot 5 + 12 a \cdot 3}{3 (2 a + 5) (2 a - 5)}$

Step 6.3

Multiply $7$ by $5$ .

$\frac{14 a + 35 + 12 a \cdot 3}{3 (2 a + 5) (2 a - 5)}$

Step 6.4

Multiply $3$ by $12$ .

$\frac{14 a + 35 + 36 a}{3 (2 a + 5) (2 a - 5)}$

Step 6.5

Add $14 a$ and $36 a$ .

$\frac{50 a + 35}{3 (2 a + 5) (2 a - 5)}$

Step 6.6

Factor $5$ out of $50 a + 35$ .

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Step 6.6.1

Factor $5$ out of $50 a$ .

$\frac{5 (10 a) + 35}{3 (2 a + 5) (2 a - 5)}$

Step 6.6.2

Factor $5$ out of $35$ .

$\frac{5 (10 a) + 5 (7)}{3 (2 a + 5) (2 a - 5)}$

Step 6.6.3

Factor $5$ out of $5 (10 a) + 5 (7)$ .

$\frac{5 (10 a + 7)}{3 (2 a + 5) (2 a - 5)}$