Simplify (a+1-6/a)/(a-1-2/a)

$\frac{a + 1 - \frac{6}{a}}{a - 1 - \frac{2}{a}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $a$ .

Multiply $\frac{a + 1 - \frac{6}{a}}{a - 1 - \frac{2}{a}}$ by $\frac{a}{a}$ .

$\frac{a}{a} \cdot \frac{a + 1 - \frac{6}{a}}{a - 1 - \frac{2}{a}}$

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $a$ .

Move the leading negative in $- \frac{6}{a}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $a$ .

Move the leading negative in $- \frac{2}{a}$ into the numerator.

$\frac{a \cdot a + a \cdot 1 - 6}{a \cdot a + a \cdot - 1 + a \frac{- 2}{a}}$

Cancel the common factor.

$\frac{a \cdot a + a \cdot 1 - 6}{a \cdot a + a \cdot - 1 + a \frac{- 2}{a}}$

Rewrite the expression.

$\frac{a \cdot a + a \cdot 1 - 6}{a \cdot a + a \cdot - 1 - 2}$

Step 4

Simplify the numerator.

Multiply $a$ by $a$ .

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

Multiply $a$ by $1$ .

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

Factor $a^{2} + a - 6$ 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 $1$ .

$- 2, 3$

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

Multiply $a$ by $a$ .

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

Move $- 1$ to the left of $a$ .

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

Rewrite $- 1 a$ as $- a$ .

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

Factor $a^{2} - a - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $a - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{a + 3}{a + 1}$

Simplify (a+1-6/a)/(a-1-2/a)

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