Finite Math Examples

$\frac{\frac{4 a + 12}{2 a - 10}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1

Cancel the common factor of $4 a + 12$ and $2 a - 10$ .

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Step 1.1

Factor $2$ out of $4 a$ .

$\frac{\frac{2 (2 a) + 12}{2 a - 10}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.2

Factor $2$ out of $12$ .

$\frac{\frac{2 (2 a) + 2 \cdot 6}{2 a - 10}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.3

Factor $2$ out of $2 (2 a) + 2 (6)$ .

$\frac{\frac{2 (2 a + 6)}{2 a - 10}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 a$ .

$\frac{\frac{2 (2 a + 6)}{2 (a) - 10}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.4.2

Factor $2$ out of $- 10$ .

$\frac{\frac{2 (2 a + 6)}{2 a + 2 \cdot - 5}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.4.3

Factor $2$ out of $2 a + 2 \cdot - 5$ .

$\frac{\frac{2 (2 a + 6)}{2 (a - 5)}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.4.4

Cancel the common factor.

$\frac{\frac{2 (2 a + 6)}{2 (a - 5)}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 1.4.5

Rewrite the expression.

$\frac{\frac{2 a + 6}{a - 5}}{\frac{2 - 9}{a^{2} - a - 20}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 a + 6}{a - 5} \cdot \frac{a^{2} - a - 20}{2 - 9}$

Step 3

Factor $2$ out of $2 a + 6$ .

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Step 3.1

Factor $2$ out of $2 a$ .

$\frac{2 (a) + 6}{a - 5} \cdot \frac{a^{2} - a - 20}{2 - 9}$

Step 3.2

Factor $2$ out of $6$ .

$\frac{2 a + 2 \cdot 3}{a - 5} \cdot \frac{a^{2} - a - 20}{2 - 9}$

Step 3.3

Factor $2$ out of $2 a + 2 \cdot 3$ .

$\frac{2 (a + 3)}{a - 5} \cdot \frac{a^{2} - a - 20}{2 - 9}$

Step 4

Factor $a^{2} - a - 20$ using the AC method.

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Step 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 4.2

Write the factored form using these integers.

$\frac{2 (a + 3)}{a - 5} \cdot \frac{(a - 5) (a + 4)}{2 - 9}$

Step 5

Simplify terms.

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Step 5.1

Subtract $9$ from $2$ .

$\frac{2 (a + 3)}{a - 5} \cdot \frac{(a - 5) (a + 4)}{- 7}$

Step 5.2

Cancel the common factor of $a - 5$ .

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Step 5.2.1

Cancel the common factor.

$\frac{2 (a + 3)}{a - 5} \cdot \frac{(a - 5) (a + 4)}{- 7}$

Step 5.2.2

Rewrite the expression.

$2 (a + 3) \frac{a + 4}{- 7}$

Step 5.3

Combine $\frac{a + 4}{- 7}$ and $2$ .

$\frac{(a + 4) \cdot 2}{- 7} (a + 3)$

Step 5.4

Simplify the expression.

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Step 5.4.1

Move $2$ to the left of $a + 4$ .

$\frac{2 \cdot (a + 4)}{- 7} (a + 3)$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{2 \cdot (a + 4)}{7} (a + 3)$

Step 5.5

Apply the distributive property.

$- \frac{2 (a + 4)}{7} a - \frac{2 (a + 4)}{7} \cdot 3$

Step 5.6

Combine $a$ and $\frac{2 (a + 4)}{7}$ .

$- \frac{a (2 (a + 4))}{7} - \frac{2 (a + 4)}{7} \cdot 3$

Step 6

Multiply $- \frac{2 (a + 4)}{7} \cdot 3$ .

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Step 6.1

Multiply $3$ by $- 1$ .

$- \frac{a (2 (a + 4))}{7} - 3 \frac{2 (a + 4)}{7}$

Step 6.2

Combine $- 3$ and $\frac{2 (a + 4)}{7}$ .

$- \frac{a (2 (a + 4))}{7} + \frac{- 3 (2 (a + 4))}{7}$

Step 6.3

Multiply $2$ by $- 3$ .

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

Step 7

Simplify terms.

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Step 7.1

Combine the numerators over the common denominator.

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

Step 7.2

Simplify each term.

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Step 7.2.1

Apply the distributive property.

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

Step 7.2.2

Multiply $2$ by $4$ .

$\frac{- a (2 a + 8) - 6 (a + 4)}{7}$

Step 7.2.3

Apply the distributive property.

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

Step 7.2.4

Rewrite using the commutative property of multiplication.

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

Step 7.2.5

Multiply $8$ by $- 1$ .

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

Step 7.2.6

Simplify each term.

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Step 7.2.6.1

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

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Step 7.2.6.1.1

Move $a$ .

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

Step 7.2.6.1.2

Multiply $a$ by $a$ .

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

Step 7.2.6.2

Multiply $- 1$ by $2$ .

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

Step 7.2.7

Apply the distributive property.

$\frac{- 2 a^{2} - 8 a - 6 a - 6 \cdot 4}{7}$

Step 7.2.8

Multiply $- 6$ by $4$ .

$\frac{- 2 a^{2} - 8 a - 6 a - 24}{7}$

Step 7.3

Subtract $6 a$ from $- 8 a$ .

$\frac{- 2 a^{2} - 14 a - 24}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $2$ out of $- 2 a^{2} - 14 a - 24$ .

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Step 8.1.1

Factor $2$ out of $- 2 a^{2}$ .

$\frac{2 (- a^{2}) - 14 a - 24}{7}$

Step 8.1.2

Factor $2$ out of $- 14 a$ .

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

Step 8.1.3

Factor $2$ out of $- 24$ .

$\frac{2 (- a^{2}) + 2 (- 7 a) + 2 (- 12)}{7}$

Step 8.1.4

Factor $2$ out of $2 (- a^{2}) + 2 (- 7 a)$ .

$\frac{2 (- a^{2} - 7 a) + 2 (- 12)}{7}$

Step 8.1.5

Factor $2$ out of $2 (- a^{2} - 7 a) + 2 (- 12)$ .

$\frac{2 (- a^{2} - 7 a - 12)}{7}$

Step 8.2

Factor by grouping.

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Step 8.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 12 = 12$ and whose sum is $b = - 7$ .

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Step 8.2.1.1

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

$\frac{2 (- a^{2} - 7 (a) - 12)}{7}$

Step 8.2.1.2

Rewrite $- 7$ as $- 3$ plus $- 4$

$\frac{2 (- a^{2} + (- 3 - 4) a - 12)}{7}$

Step 8.2.1.3

Apply the distributive property.

$\frac{2 (- a^{2} - 3 a - 4 a - 12)}{7}$

Step 8.2.2

Factor out the greatest common factor from each group.

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Step 8.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((- a^{2} - 3 a) - 4 a - 12)}{7}$

Step 8.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (a (- a - 3) + 4 (- a - 3))}{7}$

Step 8.2.3

Factor the polynomial by factoring out the greatest common factor, $- a - 3$ .

$\frac{2 ((- a - 3) (a + 4))}{7}$

$\frac{2 (- a - 3) (a + 4)}{7}$

Step 9

Simplify with factoring out.

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Step 9.1

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

$\frac{2 (- (a) - 3) (a + 4)}{7}$

Step 9.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{2 (- (a) - 1 (3)) (a + 4)}{7}$

Step 9.3

Factor $- 1$ out of $- (a) - 1 (3)$ .

$\frac{2 (- (a + 3)) (a + 4)}{7}$

Step 9.4

Simplify the expression.

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Step 9.4.1

Rewrite $- (a + 3)$ as $- 1 (a + 3)$ .

$\frac{2 (- 1 (a + 3)) (a + 4)}{7}$

Step 9.4.2

Move the negative in front of the fraction.

$- \frac{(2 (a + 3)) (a + 4)}{7}$

Step 9.4.3

Reorder factors in $- \frac{(2 (a + 3)) (a + 4)}{7}$ .

$- \frac{2 (a + 3) (a + 4)}{7}$