Finite Math Examples

$\frac{a + 2}{a^{2} - 9} - \frac{2 a}{6 a^{2} - 17 a - 3}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{a + 2}{a^{2} - 3^{2}} - \frac{2 a}{6 a^{2} - 17 a - 3}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{6 a^{2} - 17 a - 3}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 6 \cdot - 3 = - 18$ and whose sum is $b = - 17$ .

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Step 1.2.1.1

Factor $- 17$ out of $- 17 a$ .

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{6 a^{2} - 17 (a) - 3}$

Step 1.2.1.2

Rewrite $- 17$ as $1$ plus $- 18$

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{6 a^{2} + (1 - 18) a - 3}$

Step 1.2.1.3

Apply the distributive property.

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{6 a^{2} + 1 a - 18 a - 3}$

Step 1.2.1.4

Multiply $a$ by $1$ .

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{6 a^{2} + a - 18 a - 3}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{(6 a^{2} + a) - 18 a - 3}$

Step 1.2.2.2

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

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{a (6 a + 1) - 3 (6 a + 1)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $6 a + 1$ .

$\frac{a + 2}{(a + 3) (a - 3)} - \frac{2 a}{(6 a + 1) (a - 3)}$

Step 2

To write $\frac{a + 2}{(a + 3) (a - 3)}$ as a fraction with a common denominator, multiply by $\frac{6 a + 1}{6 a + 1}$ .

$\frac{a + 2}{(a + 3) (a - 3)} \cdot \frac{6 a + 1}{6 a + 1} - \frac{2 a}{(6 a + 1) (a - 3)}$

Step 3

To write $- \frac{2 a}{(6 a + 1) (a - 3)}$ as a fraction with a common denominator, multiply by $\frac{a + 3}{a + 3}$ .

$\frac{a + 2}{(a + 3) (a - 3)} \cdot \frac{6 a + 1}{6 a + 1} - \frac{2 a}{(6 a + 1) (a - 3)} \cdot \frac{a + 3}{a + 3}$

Step 4

Write each expression with a common denominator of $(a + 3) (a - 3) (6 a + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a + 2}{(a + 3) (a - 3)}$ by $\frac{6 a + 1}{6 a + 1}$ .

$\frac{(a + 2) (6 a + 1)}{(a + 3) (a - 3) (6 a + 1)} - \frac{2 a}{(6 a + 1) (a - 3)} \cdot \frac{a + 3}{a + 3}$

Step 4.2

Multiply $\frac{2 a}{(6 a + 1) (a - 3)}$ by $\frac{a + 3}{a + 3}$ .

$\frac{(a + 2) (6 a + 1)}{(a + 3) (a - 3) (6 a + 1)} - \frac{2 a (a + 3)}{(6 a + 1) (a - 3) (a + 3)}$

Step 4.3

Reorder the factors of $(a + 3) (a - 3) (6 a + 1)$ .

$\frac{(a + 2) (6 a + 1)}{(6 a + 1) (a + 3) (a - 3)} - \frac{2 a (a + 3)}{(6 a + 1) (a - 3) (a + 3)}$

Step 4.4

Reorder the factors of $(6 a + 1) (a - 3) (a + 3)$ .

$\frac{(a + 2) (6 a + 1)}{(6 a + 1) (a + 3) (a - 3)} - \frac{2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(a + 2) (6 a + 1) - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(a + 2) (6 a + 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{a (6 a + 1) + 2 (6 a + 1) - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.1.2

Apply the distributive property.

$\frac{a (6 a) + a \cdot 1 + 2 (6 a + 1) - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.1.3

Apply the distributive property.

$\frac{a (6 a) + a \cdot 1 + 2 (6 a) + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 a \cdot a + a \cdot 1 + 2 (6 a) + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.1.2

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

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Step 6.2.1.2.1

Move $a$ .

$\frac{6 (a \cdot a) + a \cdot 1 + 2 (6 a) + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.1.2.2

Multiply $a$ by $a$ .

$\frac{6 a^{2} + a \cdot 1 + 2 (6 a) + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.1.3

Multiply $a$ by $1$ .

$\frac{6 a^{2} + a + 2 (6 a) + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.1.4

Multiply $6$ by $2$ .

$\frac{6 a^{2} + a + 12 a + 2 \cdot 1 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.1.5

Multiply $2$ by $1$ .

$\frac{6 a^{2} + a + 12 a + 2 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.2.2

Add $a$ and $12 a$ .

$\frac{6 a^{2} + 13 a + 2 - 2 a (a + 3)}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.3

Apply the distributive property.

$\frac{6 a^{2} + 13 a + 2 - 2 a \cdot a - 2 a \cdot 3}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.4

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

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Step 6.4.1

Move $a$ .

$\frac{6 a^{2} + 13 a + 2 - 2 (a \cdot a) - 2 a \cdot 3}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.4.2

Multiply $a$ by $a$ .

$\frac{6 a^{2} + 13 a + 2 - 2 a^{2} - 2 a \cdot 3}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.5

Multiply $3$ by $- 2$ .

$\frac{6 a^{2} + 13 a + 2 - 2 a^{2} - 6 a}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.6

Subtract $2 a^{2}$ from $6 a^{2}$ .

$\frac{4 a^{2} + 13 a + 2 - 6 a}{(6 a + 1) (a + 3) (a - 3)}$

Step 6.7

Subtract $6 a$ from $13 a$ .

$\frac{4 a^{2} + 7 a + 2}{(6 a + 1) (a + 3) (a - 3)}$