Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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.

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Step 3.1

Cancel the common factor of $a$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $a$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $a$ by $a$ .

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

Step 4.2

Multiply $a$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 4.3.2

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

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Step 5.1

Multiply $a$ by $a$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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Step 5.4.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 5.4.2

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $a - 2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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