Simplify (a^2-2a-3)/(a^2-9a+18)-(a^2-5a-6)/(a^2+9a+8)

$\frac{a^{2} - 2 a - 3}{a^{2} - 9 a + 18} - \frac{a^{2} - 5 a - 6}{a^{2} + 9 a + 8}$

Step 1

Simplify each term.

Factor $a^{2} - 2 a - 3$ using the AC method.

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$\frac{(a - 3) (a + 1)}{a^{2} - 9 a + 18} - \frac{a^{2} - 5 a - 6}{a^{2} + 9 a + 8}$

Factor $a^{2} - 9 a + 18$ using the AC method.

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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$\frac{(a - 3) (a + 1)}{(a - 6) (a - 3)} - \frac{a^{2} - 5 a - 6}{a^{2} + 9 a + 8}$

Cancel the common factor of $a - 3$ .

Cancel the common factor.

$\frac{(a - 3) (a + 1)}{(a - 6) (a - 3)} - \frac{a^{2} - 5 a - 6}{a^{2} + 9 a + 8}$

Rewrite the expression.

$\frac{a + 1}{a - 6} - \frac{a^{2} - 5 a - 6}{a^{2} + 9 a + 8}$

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

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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{a + 1}{a - 6} - \frac{(a - 6) (a + 1)}{a^{2} + 9 a + 8}$

Factor $a^{2} + 9 a + 8$ using the AC method.

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 $9$ .

$1, 8$

Write the factored form using these integers.

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

Cancel the common factor of $a + 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{a + 1}{a - 6} - \frac{a - 6}{a + 8}$

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(a - 6) (a + 8)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a + 1}{a - 6}$ by $\frac{a + 8}{a + 8}$ .

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

Multiply $\frac{a - 6}{a + 8}$ by $\frac{a - 6}{a - 6}$ .

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

Reorder the factors of $(a - 6) (a + 8)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Expand $(a + 1) (a + 8)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Move $8$ to the left of $a$ .

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

Multiply $a$ by $1$ .

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

Multiply $8$ by $1$ .

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

Add $8 a$ and $a$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 6$ .

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

Expand $(- a + 6) (a - 6)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$\frac{a^{2} + 9 a + 8 - a \cdot a - a \cdot - 6 + 6 (a - 6)}{(a + 8) (a - 6)}$

Apply the distributive property.

$\frac{a^{2} + 9 a + 8 - a \cdot a - a \cdot - 6 + 6 a + 6 \cdot - 6}{(a + 8) (a - 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{a^{2} + 9 a + 8 - (a \cdot a) - a \cdot - 6 + 6 a + 6 \cdot - 6}{(a + 8) (a - 6)}$

Multiply $a$ by $a$ .

$\frac{a^{2} + 9 a + 8 - a^{2} - a \cdot - 6 + 6 a + 6 \cdot - 6}{(a + 8) (a - 6)}$

Multiply $- 6$ by $- 1$ .

$\frac{a^{2} + 9 a + 8 - a^{2} + 6 a + 6 a + 6 \cdot - 6}{(a + 8) (a - 6)}$

Multiply $6$ by $- 6$ .

$\frac{a^{2} + 9 a + 8 - a^{2} + 6 a + 6 a - 36}{(a + 8) (a - 6)}$

Add $6 a$ and $6 a$ .

$\frac{a^{2} + 9 a + 8 - a^{2} + 12 a - 36}{(a + 8) (a - 6)}$

Subtract $a^{2}$ from $a^{2}$ .

$\frac{9 a + 8 + 0 + 12 a - 36}{(a + 8) (a - 6)}$

Add $9 a$ and $0$ .

$\frac{9 a + 8 + 12 a - 36}{(a + 8) (a - 6)}$

Add $9 a$ and $12 a$ .

$\frac{21 a + 8 - 36}{(a + 8) (a - 6)}$

Subtract $36$ from $8$ .

$\frac{21 a - 28}{(a + 8) (a - 6)}$

Factor $7$ out of $21 a - 28$ .

Factor $7$ out of $21 a$ .

$\frac{7 (3 a) - 28}{(a + 8) (a - 6)}$

Factor $7$ out of $- 28$ .

$\frac{7 (3 a) + 7 (- 4)}{(a + 8) (a - 6)}$

Factor $7$ out of $7 (3 a) + 7 (- 4)$ .

$\frac{7 (3 a - 4)}{(a + 8) (a - 6)}$

Simplify (a^2-2a-3)/(a^2-9a+18)-(a^2-5a-6)/(a^2+9a+8)

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