Basic Math Examples

$a^{2} - a - \frac{12}{a^{2} - 2 a - 8}$

Step 1

Find the common denominator.

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Step 1.1

Write $a^{2}$ as a fraction with denominator $1$ .

$\frac{a^{2}}{1} - a - \frac{12}{a^{2} - 2 a - 8}$

Step 1.2

Multiply $\frac{a^{2}}{1}$ by $\frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8}$ .

$\frac{a^{2}}{1} \cdot \frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8} - a - \frac{12}{a^{2} - 2 a - 8}$

Step 1.3

Multiply $\frac{a^{2}}{1}$ by $\frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8}$ .

$\frac{a^{2} (a^{2} - 2 a - 8)}{a^{2} - 2 a - 8} - a - \frac{12}{a^{2} - 2 a - 8}$

Step 1.4

Write $- a$ as a fraction with denominator $1$ .

$\frac{a^{2} (a^{2} - 2 a - 8)}{a^{2} - 2 a - 8} + \frac{- a}{1} - \frac{12}{a^{2} - 2 a - 8}$

Step 1.5

Multiply $\frac{- a}{1}$ by $\frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8}$ .

$\frac{a^{2} (a^{2} - 2 a - 8)}{a^{2} - 2 a - 8} + \frac{- a}{1} \cdot \frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8} - \frac{12}{a^{2} - 2 a - 8}$

Step 1.6

Multiply $\frac{- a}{1}$ by $\frac{a^{2} - 2 a - 8}{a^{2} - 2 a - 8}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{a^{2} (a^{2} - 2 a - 8) - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

$\frac{a^{2} a^{2} + a^{2} (- 2 a) + a^{2} \cdot - 8 - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

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Step 2.2.2.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a^{2 + 2} + a^{2} (- 2 a) + a^{2} \cdot - 8 - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.2.1.2

Add $2$ and $2$ .

$\frac{a^{4} + a^{2} (- 2 a) + a^{2} \cdot - 8 - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.2.2

Rewrite using the commutative property of multiplication.

$\frac{a^{4} - 2 a^{2} a + a^{2} \cdot - 8 - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.2.3

Move $- 8$ to the left of $a^{2}$ .

$\frac{a^{4} - 2 a^{2} a - 8 \cdot a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.3

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 2.2.3.1

Move $a$ .

$\frac{a^{4} - 2 (a \cdot a^{2}) - 8 \cdot a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.3.2

Multiply $a$ by $a^{2}$ .

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Step 2.2.3.2.1

Raise $a$ to the power of $1$ .

$\frac{a^{4} - 2 (a^{1} a^{2}) - 8 \cdot a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a^{4} - 2 a^{1 + 2} - 8 \cdot a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.3.3

Add $1$ and $2$ .

$\frac{a^{4} - 2 a^{3} - 8 \cdot a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a (a^{2} - 2 a - 8) - 12}{a^{2} - 2 a - 8}$

Step 2.2.4

Apply the distributive property.

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a \cdot a^{2} - a (- 2 a) - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 2.2.5.1.1

Move $a^{2}$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - (a^{2} a) - a (- 2 a) - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5.1.2

Multiply $a^{2}$ by $a$ .

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Step 2.2.5.1.2.1

Raise $a$ to the power of $1$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - (a^{2} a^{1}) - a (- 2 a) - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{2 + 1} - a (- 2 a) - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5.1.3

Add $2$ and $1$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} - a (- 2 a) - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5.2

Rewrite using the commutative property of multiplication.

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} - 1 \cdot - 2 a \cdot a - a \cdot - 8 - 12}{a^{2} - 2 a - 8}$

Step 2.2.5.3

Multiply $- 8$ by $- 1$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} - 1 \cdot - 2 a \cdot a + 8 a - 12}{a^{2} - 2 a - 8}$

Step 2.2.6

Simplify each term.

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Step 2.2.6.1

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

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Step 2.2.6.1.1

Move $a$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} - 1 \cdot - 2 (a \cdot a) + 8 a - 12}{a^{2} - 2 a - 8}$

Step 2.2.6.1.2

Multiply $a$ by $a$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} - 1 \cdot - 2 a^{2} + 8 a - 12}{a^{2} - 2 a - 8}$

Step 2.2.6.2

Multiply $- 1$ by $- 2$ .

$\frac{a^{4} - 2 a^{3} - 8 a^{2} - a^{3} + 2 a^{2} + 8 a - 12}{a^{2} - 2 a - 8}$

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Subtract $a^{3}$ from $- 2 a^{3}$ .

$\frac{a^{4} - 3 a^{3} - 8 a^{2} + 2 a^{2} + 8 a - 12}{a^{2} - 2 a - 8}$

Step 2.3.2

Add $- 8 a^{2}$ and $2 a^{2}$ .

$\frac{a^{4} - 3 a^{3} - 6 a^{2} + 8 a - 12}{a^{2} - 2 a - 8}$

Step 3

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

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Step 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.2

Write the factored form using these integers.

$\frac{a^{4} - 3 a^{3} - 6 a^{2} + 8 a - 12}{(a - 4) (a + 2)}$