Basic Math Examples

$\frac{6}{a} + \frac{2}{a - 1} - \frac{7}{a - 4}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{6}{a}$ by $\frac{(a - 1) (a - 4)}{(a - 1) (a - 4)}$ .

$\frac{6}{a} \cdot \frac{(a - 1) (a - 4)}{(a - 1) (a - 4)} + \frac{2}{a - 1} - \frac{7}{a - 4}$

Step 1.2

Multiply $\frac{6}{a}$ by $\frac{(a - 1) (a - 4)}{(a - 1) (a - 4)}$ .

$\frac{6 ((a - 1) (a - 4))}{a ((a - 1) (a - 4))} + \frac{2}{a - 1} - \frac{7}{a - 4}$

Step 1.3

Multiply $\frac{2}{a - 1}$ by $\frac{a (a - 4)}{a (a - 4)}$ .

$\frac{6 ((a - 1) (a - 4))}{a ((a - 1) (a - 4))} + \frac{2}{a - 1} \cdot \frac{a (a - 4)}{a (a - 4)} - \frac{7}{a - 4}$

Step 1.4

Multiply $\frac{2}{a - 1}$ by $\frac{a (a - 4)}{a (a - 4)}$ .

$\frac{6 ((a - 1) (a - 4))}{a ((a - 1) (a - 4))} + \frac{2 (a (a - 4))}{(a - 1) (a (a - 4))} - \frac{7}{a - 4}$

Step 1.5

Multiply $\frac{7}{a - 4}$ by $\frac{a (a - 1)}{a (a - 1)}$ .

$\frac{6 ((a - 1) (a - 4))}{a ((a - 1) (a - 4))} + \frac{2 (a (a - 4))}{(a - 1) (a (a - 4))} - (\frac{7}{a - 4} \cdot \frac{a (a - 1)}{a (a - 1)})$

Step 1.6

Multiply $\frac{7}{a - 4}$ by $\frac{a (a - 1)}{a (a - 1)}$ .

$\frac{6 ((a - 1) (a - 4))}{a ((a - 1) (a - 4))} + \frac{2 (a (a - 4))}{(a - 1) (a (a - 4))} - \frac{7 (a (a - 1))}{(a - 4) (a (a - 1))}$

Step 1.7

Reorder the factors of $a ((a - 1) (a - 4))$ .

$\frac{6 ((a - 1) (a - 4))}{a (a - 1) (a - 4)} + \frac{2 (a (a - 4))}{(a - 1) (a (a - 4))} - \frac{7 (a (a - 1))}{(a - 4) (a (a - 1))}$

Step 1.8

Reorder the factors of $(a - 1) (a (a - 4))$ .

$\frac{6 ((a - 1) (a - 4))}{a (a - 1) (a - 4)} + \frac{2 (a (a - 4))}{a (a - 1) (a - 4)} - \frac{7 (a (a - 1))}{(a - 4) (a (a - 1))}$

Step 1.9

Reorder the factors of $(a - 4) (a (a - 1))$ .

$\frac{6 ((a - 1) (a - 4))}{a (a - 1) (a - 4)} + \frac{2 (a (a - 4))}{a (a - 1) (a - 4)} - \frac{7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 2

Combine the numerators over the common denominator.

$\frac{6 ((a - 1) (a - 4)) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3

Simplify each term.

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Step 3.1

Expand $(a - 1) (a - 4)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{6 (a (a - 4) - 1 (a - 4)) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.1.2

Apply the distributive property.

$\frac{6 (a \cdot a + a \cdot - 4 - 1 (a - 4)) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.1.3

Apply the distributive property.

$\frac{6 (a \cdot a + a \cdot - 4 - 1 a - 1 \cdot - 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $a$ by $a$ .

$\frac{6 (a^{2} + a \cdot - 4 - 1 a - 1 \cdot - 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.2.1.2

Move $- 4$ to the left of $a$ .

$\frac{6 (a^{2} - 4 \cdot a - 1 a - 1 \cdot - 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.2.1.3

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

$\frac{6 (a^{2} - 4 a - a - 1 \cdot - 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.2.1.4

Multiply $- 1$ by $- 4$ .

$\frac{6 (a^{2} - 4 a - a + 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.2.2

Subtract $a$ from $- 4 a$ .

$\frac{6 (a^{2} - 5 a + 4) + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.3

Apply the distributive property.

$\frac{6 a^{2} + 6 (- 5 a) + 6 \cdot 4 + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $- 5$ by $6$ .

$\frac{6 a^{2} - 30 a + 6 \cdot 4 + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.4.2

Multiply $6$ by $4$ .

$\frac{6 a^{2} - 30 a + 24 + 2 (a (a - 4)) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.5

Apply the distributive property.

$\frac{6 a^{2} - 30 a + 24 + 2 (a \cdot a + a \cdot - 4) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.6

Multiply $a$ by $a$ .

$\frac{6 a^{2} - 30 a + 24 + 2 (a^{2} + a \cdot - 4) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.7

Move $- 4$ to the left of $a$ .

$\frac{6 a^{2} - 30 a + 24 + 2 (a^{2} - 4 \cdot a) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.8

Apply the distributive property.

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} + 2 (- 4 a) - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.9

Multiply $- 4$ by $2$ .

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 (a (a - 1))}{a (a - 1) (a - 4)}$

Step 3.10

Apply the distributive property.

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 (a \cdot a + a \cdot - 1)}{a (a - 1) (a - 4)}$

Step 3.11

Multiply $a$ by $a$ .

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 (a^{2} + a \cdot - 1)}{a (a - 1) (a - 4)}$

Step 3.12

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

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 (a^{2} - 1 \cdot a)}{a (a - 1) (a - 4)}$

Step 3.13

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

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 (a^{2} - a)}{a (a - 1) (a - 4)}$

Step 3.14

Apply the distributive property.

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 a^{2} - 7 (- a)}{a (a - 1) (a - 4)}$

Step 3.15

Multiply $- 1$ by $- 7$ .

$\frac{6 a^{2} - 30 a + 24 + 2 a^{2} - 8 a - 7 a^{2} + 7 a}{a (a - 1) (a - 4)}$

Step 4

Simplify by adding terms.

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Step 4.1

Add $6 a^{2}$ and $2 a^{2}$ .

$\frac{8 a^{2} - 30 a + 24 - 8 a - 7 a^{2} + 7 a}{a (a - 1) (a - 4)}$

Step 4.2

Subtract $7 a^{2}$ from $8 a^{2}$ .

$\frac{a^{2} - 30 a + 24 - 8 a + 7 a}{a (a - 1) (a - 4)}$

Step 4.3

Subtract $8 a$ from $- 30 a$ .

$\frac{a^{2} - 38 a + 24 + 7 a}{a (a - 1) (a - 4)}$

Step 4.4

Add $- 38 a$ and $7 a$ .

$\frac{a^{2} - 31 a + 24}{a (a - 1) (a - 4)}$