Basic Math Examples

$\frac{6}{a^{2} + 8 a + 15} + \frac{- 12}{a^{2} + 6 a + 5}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} + 8 a + 15$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $8$ .

$3, 5$

Step 1.1.2

Write the factored form using these integers.

$\frac{6}{(a + 3) (a + 5)} + \frac{- 12}{a^{2} + 6 a + 5}$

Step 1.2

Factor $a^{2} + 6 a + 5$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $6$ .

$1, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{6}{(a + 3) (a + 5)} + \frac{- 12}{(a + 1) (a + 5)}$

Step 1.3

Move the negative in front of the fraction.

$\frac{6}{(a + 3) (a + 5)} - \frac{12}{(a + 1) (a + 5)}$

Step 2

To write $\frac{6}{(a + 3) (a + 5)}$ as a fraction with a common denominator, multiply by $\frac{a + 1}{a + 1}$ .

$\frac{6}{(a + 3) (a + 5)} \cdot \frac{a + 1}{a + 1} - \frac{12}{(a + 1) (a + 5)}$

Step 3

To write $- \frac{12}{(a + 1) (a + 5)}$ as a fraction with a common denominator, multiply by $\frac{a + 3}{a + 3}$ .

$\frac{6}{(a + 3) (a + 5)} \cdot \frac{a + 1}{a + 1} - \frac{12}{(a + 1) (a + 5)} \cdot \frac{a + 3}{a + 3}$

Step 4

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

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Step 4.1

Multiply $\frac{6}{(a + 3) (a + 5)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{6 (a + 1)}{(a + 3) (a + 5) (a + 1)} - \frac{12}{(a + 1) (a + 5)} \cdot \frac{a + 3}{a + 3}$

Step 4.2

Multiply $\frac{12}{(a + 1) (a + 5)}$ by $\frac{a + 3}{a + 3}$ .

$\frac{6 (a + 1)}{(a + 3) (a + 5) (a + 1)} - \frac{12 (a + 3)}{(a + 1) (a + 5) (a + 3)}$

Step 4.3

Reorder the factors of $(a + 3) (a + 5) (a + 1)$ .

$\frac{6 (a + 1)}{(a + 1) (a + 3) (a + 5)} - \frac{12 (a + 3)}{(a + 1) (a + 5) (a + 3)}$

Step 4.4

Reorder the factors of $(a + 1) (a + 5) (a + 3)$ .

$\frac{6 (a + 1)}{(a + 1) (a + 3) (a + 5)} - \frac{12 (a + 3)}{(a + 1) (a + 3) (a + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 (a + 1) - 12 (a + 3)}{(a + 1) (a + 3) (a + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $6$ out of $6 (a + 1) - 12 (a + 3)$ .

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Step 6.1.1

Factor $6$ out of $- 12 (a + 3)$ .

$\frac{6 (a + 1) + 6 (- 2 (a + 3))}{(a + 1) (a + 3) (a + 5)}$

Step 6.1.2

Factor $6$ out of $6 (a + 1) + 6 (- 2 (a + 3))$ .

$\frac{6 (a + 1 - 2 (a + 3))}{(a + 1) (a + 3) (a + 5)}$

Step 6.2

Apply the distributive property.

$\frac{6 (a + 1 - 2 a - 2 \cdot 3)}{(a + 1) (a + 3) (a + 5)}$

Step 6.3

Multiply $- 2$ by $3$ .

$\frac{6 (a + 1 - 2 a - 6)}{(a + 1) (a + 3) (a + 5)}$

Step 6.4

Subtract $2 a$ from $a$ .

$\frac{6 (- a + 1 - 6)}{(a + 1) (a + 3) (a + 5)}$

Step 6.5

Subtract $6$ from $1$ .

$\frac{6 (- a - 5)}{(a + 1) (a + 3) (a + 5)}$

Step 7

Cancel the common factor of $- a - 5$ and $a + 5$ .

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Step 7.1

Factor $- 1$ out of $- a$ .

$\frac{6 (- (a) - 5)}{(a + 1) (a + 3) (a + 5)}$

Step 7.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{6 (- (a) - 1 (5))}{(a + 1) (a + 3) (a + 5)}$

Step 7.3

Factor $- 1$ out of $- (a) - 1 (5)$ .

$\frac{6 (- (a + 5))}{(a + 1) (a + 3) (a + 5)}$

Step 7.4

Rewrite $- (a + 5)$ as $- 1 (a + 5)$ .

$\frac{6 (- 1 (a + 5))}{(a + 1) (a + 3) (a + 5)}$

Step 7.5

Cancel the common factor.

$\frac{6 (- 1 (a + 5))}{(a + 1) (a + 3) (a + 5)}$

Step 7.6

Rewrite the expression.

$\frac{6 \cdot (- 1)}{(a + 1) (a + 3)}$

Step 8

Multiply $6$ by $- 1$ .

$\frac{- 6}{(a + 1) (a + 3)}$

Step 9

Move the negative in front of the fraction.

$- \frac{6}{(a + 1) (a + 3)}$