Trigonometry Examples

$\frac{7 a}{a^{2} + 9 a + 8} - \frac{6 a}{a^{2} + 8 a + 7}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} + 9 a + 8$ 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 $8$ and whose sum is $9$ .

$1, 8$

Step 1.1.2

Write the factored form using these integers.

$\frac{7 a}{(a + 1) (a + 8)} - \frac{6 a}{a^{2} + 8 a + 7}$

Step 1.2

Factor $a^{2} + 8 a + 7$ 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 $7$ and whose sum is $8$ .

$1, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{7 a}{(a + 1) (a + 8)} - \frac{6 a}{(a + 1) (a + 7)}$

Step 2

To write $\frac{7 a}{(a + 1) (a + 8)}$ as a fraction with a common denominator, multiply by $\frac{a + 7}{a + 7}$ .

$\frac{7 a}{(a + 1) (a + 8)} \cdot \frac{a + 7}{a + 7} - \frac{6 a}{(a + 1) (a + 7)}$

Step 3

To write $- \frac{6 a}{(a + 1) (a + 7)}$ as a fraction with a common denominator, multiply by $\frac{a + 8}{a + 8}$ .

$\frac{7 a}{(a + 1) (a + 8)} \cdot \frac{a + 7}{a + 7} - \frac{6 a}{(a + 1) (a + 7)} \cdot \frac{a + 8}{a + 8}$

Step 4

Write each expression with a common denominator of $(a + 1) (a + 8) (a + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{7 a}{(a + 1) (a + 8)}$ by $\frac{a + 7}{a + 7}$ .

$\frac{7 a (a + 7)}{(a + 1) (a + 8) (a + 7)} - \frac{6 a}{(a + 1) (a + 7)} \cdot \frac{a + 8}{a + 8}$

Step 4.2

Multiply $\frac{6 a}{(a + 1) (a + 7)}$ by $\frac{a + 8}{a + 8}$ .

$\frac{7 a (a + 7)}{(a + 1) (a + 8) (a + 7)} - \frac{6 a (a + 8)}{(a + 1) (a + 7) (a + 8)}$

Step 4.3

Reorder the factors of $(a + 1) (a + 8) (a + 7)$ .

$\frac{7 a (a + 7)}{(a + 1) (a + 7) (a + 8)} - \frac{6 a (a + 8)}{(a + 1) (a + 7) (a + 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{7 a (a + 7) - 6 a (a + 8)}{(a + 1) (a + 7) (a + 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $a$ out of $7 a (a + 7) - 6 a (a + 8)$ .

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Step 6.1.1

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

$\frac{a (7 (a + 7)) - 6 a (a + 8)}{(a + 1) (a + 7) (a + 8)}$

Step 6.1.2

Factor $a$ out of $- 6 a (a + 8)$ .

$\frac{a (7 (a + 7)) + a (- 6 (a + 8))}{(a + 1) (a + 7) (a + 8)}$

Step 6.1.3

Factor $a$ out of $a (7 (a + 7)) + a (- 6 (a + 8))$ .

$\frac{a (7 (a + 7) - 6 (a + 8))}{(a + 1) (a + 7) (a + 8)}$

Step 6.2

Apply the distributive property.

$\frac{a (7 a + 7 \cdot 7 - 6 (a + 8))}{(a + 1) (a + 7) (a + 8)}$

Step 6.3

Multiply $7$ by $7$ .

$\frac{a (7 a + 49 - 6 (a + 8))}{(a + 1) (a + 7) (a + 8)}$

Step 6.4

Apply the distributive property.

$\frac{a (7 a + 49 - 6 a - 6 \cdot 8)}{(a + 1) (a + 7) (a + 8)}$

Step 6.5

Multiply $- 6$ by $8$ .

$\frac{a (7 a + 49 - 6 a - 48)}{(a + 1) (a + 7) (a + 8)}$

Step 6.6

Subtract $6 a$ from $7 a$ .

$\frac{a (a + 49 - 48)}{(a + 1) (a + 7) (a + 8)}$

Step 6.7

Subtract $48$ from $49$ .

$\frac{a (a + 1)}{(a + 1) (a + 7) (a + 8)}$

Step 7

Cancel the common factor of $a + 1$ .

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Step 7.1

Cancel the common factor.

$\frac{a (a + 1)}{(a + 1) (a + 7) (a + 8)}$

Step 7.2

Rewrite the expression.

$\frac{a}{(a + 7) (a + 8)}$