Basic Math Examples

$\frac{a}{a^{2} - 2 a - 63} - \frac{8}{a + 7}$

Step 1

Factor $a^{2} - 2 a - 63$ using the AC method.

Tap for more steps...

Step 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 $- 63$ and whose sum is $- 2$ .

$- 9, 7$

Step 1.2

Write the factored form using these integers.

$\frac{a}{(a - 9) (a + 7)} - \frac{8}{a + 7}$

Step 2

To write $- \frac{8}{a + 7}$ as a fraction with a common denominator, multiply by $\frac{a - 9}{a - 9}$ .

$\frac{a}{(a - 9) (a + 7)} - \frac{8}{a + 7} \cdot \frac{a - 9}{a - 9}$

Step 3

Write each expression with a common denominator of $(a - 9) (a + 7)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{8}{a + 7}$ by $\frac{a - 9}{a - 9}$ .

$\frac{a}{(a - 9) (a + 7)} - \frac{8 (a - 9)}{(a + 7) (a - 9)}$

Step 3.2

Reorder the factors of $(a - 9) (a + 7)$ .

$\frac{a}{(a + 7) (a - 9)} - \frac{8 (a - 9)}{(a + 7) (a - 9)}$

Step 4

Combine the numerators over the common denominator.

$\frac{a - 8 (a - 9)}{(a + 7) (a - 9)}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Apply the distributive property.

$\frac{a - 8 a - 8 \cdot - 9}{(a + 7) (a - 9)}$

Step 5.2

Multiply $- 8$ by $- 9$ .

$\frac{a - 8 a + 72}{(a + 7) (a - 9)}$

Step 5.3

Subtract $8 a$ from $a$ .

$\frac{- 7 a + 72}{(a + 7) (a - 9)}$

Step 6

Simplify with factoring out.

Tap for more steps...

Step 6.1

Factor $- 1$ out of $- 7 a$ .

$\frac{- (7 a) + 72}{(a + 7) (a - 9)}$

Step 6.2

Rewrite $72$ as $- 1 (- 72)$ .

$\frac{- (7 a) - 1 (- 72)}{(a + 7) (a - 9)}$

Step 6.3

Factor $- 1$ out of $- (7 a) - 1 (- 72)$ .

$\frac{- (7 a - 72)}{(a + 7) (a - 9)}$

Step 6.4

Simplify the expression.

Tap for more steps...

Step 6.4.1

Rewrite $- (7 a - 72)$ as $- 1 (7 a - 72)$ .

$\frac{- 1 (7 a - 72)}{(a + 7) (a - 9)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{7 a - 72}{(a + 7) (a - 9)}$