Basic Math Examples

$\frac{a - 2}{6 a^{2} - 7 a - 5} \div \frac{2 a}{3 a^{2} - 5 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{a - 2}{6 a^{2} - 7 a - 5} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = - 7$ .

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Step 1.2.1.1

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

$\frac{a - 2}{6 a^{2} - 7 (a) - 5} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.2.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$\frac{a - 2}{6 a^{2} + (3 - 10) a - 5} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.2.1.3

Apply the distributive property.

$\frac{a - 2}{6 a^{2} + 3 a - 10 a - 5} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

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.

$\frac{a - 2}{(6 a^{2} + 3 a) - 10 a - 5} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.2.2.2

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

$\frac{a - 2}{3 a (2 a + 1) - 5 (2 a + 1)} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 a + 1$ .

$\frac{a - 2}{(2 a + 1) (3 a - 5)} \cdot \frac{3 a^{2} - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.3

Factor $a$ out of $3 a^{2} - 5 a$ .

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Step 1.3.1

Factor $a$ out of $3 a^{2}$ .

$\frac{a - 2}{(2 a + 1) (3 a - 5)} \cdot \frac{a (3 a) - 5 a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.3.2

Factor $a$ out of $- 5 a$ .

$\frac{a - 2}{(2 a + 1) (3 a - 5)} \cdot \frac{a (3 a) + a \cdot - 5}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.3.3

Factor $a$ out of $a (3 a) + a \cdot - 5$ .

$\frac{a - 2}{(2 a + 1) (3 a - 5)} \cdot \frac{a (3 a - 5)}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.4

Cancel the common factor of $3 a - 5$ .

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Step 1.4.1

Factor $3 a - 5$ out of $(2 a + 1) (3 a - 5)$ .

$\frac{a - 2}{(3 a - 5) (2 a + 1)} \cdot \frac{a (3 a - 5)}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.4.2

Factor $3 a - 5$ out of $a (3 a - 5)$ .

$\frac{a - 2}{(3 a - 5) (2 a + 1)} \cdot \frac{(3 a - 5) a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.4.3

Cancel the common factor.

$\frac{a - 2}{(3 a - 5) (2 a + 1)} \cdot \frac{(3 a - 5) a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.4.4

Rewrite the expression.

$\frac{a - 2}{2 a + 1} \cdot \frac{a}{2 a} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.5

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

$\frac{(a - 2) a}{(2 a + 1) (2 a)} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.6

Cancel the common factor of $a$ .

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Step 1.6.1

Cancel the common factor.

$\frac{(a - 2) a}{(2 a + 1) (2 a)} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.6.2

Rewrite the expression.

$\frac{a - 2}{(2 a + 1) \cdot (2)} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.7

Move $2$ to the left of $2 a + 1$ .

$\frac{a - 2}{2 \cdot (2 a + 1)} - \frac{3 a + 2}{2 a^{2} + 11 a + 5}$

Step 1.8

Factor by grouping.

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Step 1.8.1

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 $a \cdot c = 2 \cdot 5 = 10$ and whose sum is $b = 11$ .

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Step 1.8.1.1

Factor $11$ out of $11 a$ .

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{2 a^{2} + 11 (a) + 5}$

Step 1.8.1.2

Rewrite $11$ as $1$ plus $10$

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{2 a^{2} + (1 + 10) a + 5}$

Step 1.8.1.3

Apply the distributive property.

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{2 a^{2} + 1 a + 10 a + 5}$

Step 1.8.1.4

Multiply $a$ by $1$ .

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{2 a^{2} + a + 10 a + 5}$

Step 1.8.2

Factor out the greatest common factor from each group.

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Step 1.8.2.1

Group the first two terms and the last two terms.

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{(2 a^{2} + a) + 10 a + 5}$

Step 1.8.2.2

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

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{a (2 a + 1) + 5 (2 a + 1)}$

Step 1.8.3

Factor the polynomial by factoring out the greatest common factor, $2 a + 1$ .

$\frac{a - 2}{2 (2 a + 1)} - \frac{3 a + 2}{(2 a + 1) (a + 5)}$

Step 2

To write $\frac{a - 2}{2 (2 a + 1)}$ as a fraction with a common denominator, multiply by $\frac{a + 5}{a + 5}$ .

$\frac{a - 2}{2 (2 a + 1)} \cdot \frac{a + 5}{a + 5} - \frac{3 a + 2}{(2 a + 1) (a + 5)}$

Step 3

To write $- \frac{3 a + 2}{(2 a + 1) (a + 5)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a - 2}{2 (2 a + 1)} \cdot \frac{a + 5}{a + 5} - \frac{3 a + 2}{(2 a + 1) (a + 5)} \cdot \frac{2}{2}$

Step 4

Write each expression with a common denominator of $2 (2 a + 1) (a + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a - 2}{2 (2 a + 1)}$ by $\frac{a + 5}{a + 5}$ .

$\frac{(a - 2) (a + 5)}{2 (2 a + 1) (a + 5)} - \frac{3 a + 2}{(2 a + 1) (a + 5)} \cdot \frac{2}{2}$

Step 4.2

Multiply $\frac{3 a + 2}{(2 a + 1) (a + 5)}$ by $\frac{2}{2}$ .

$\frac{(a - 2) (a + 5)}{2 (2 a + 1) (a + 5)} - \frac{(3 a + 2) \cdot 2}{(2 a + 1) (a + 5) \cdot 2}$

Step 4.3

Reorder the factors of $(2 a + 1) (a + 5) \cdot 2$ .

$\frac{(a - 2) (a + 5)}{2 (2 a + 1) (a + 5)} - \frac{(3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(a - 2) (a + 5) - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(a - 2) (a + 5)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{a (a + 5) - 2 (a + 5) - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.1.2

Apply the distributive property.

$\frac{a \cdot a + a \cdot 5 - 2 (a + 5) - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.1.3

Apply the distributive property.

$\frac{a \cdot a + a \cdot 5 - 2 a - 2 \cdot 5 - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

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

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot 5 - 2 a - 2 \cdot 5 - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.2.1.2

Move $5$ to the left of $a$ .

$\frac{a^{2} + 5 \cdot a - 2 a - 2 \cdot 5 - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.2.1.3

Multiply $- 2$ by $5$ .

$\frac{a^{2} + 5 a - 2 a - 10 - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.2.2

Subtract $2 a$ from $5 a$ .

$\frac{a^{2} + 3 a - 10 - (3 a + 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.3

Apply the distributive property.

$\frac{a^{2} + 3 a - 10 + (- (3 a) - 1 \cdot 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.4

Multiply $3$ by $- 1$ .

$\frac{a^{2} + 3 a - 10 + (- 3 a - 1 \cdot 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.5

Multiply $- 1$ by $2$ .

$\frac{a^{2} + 3 a - 10 + (- 3 a - 2) \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.6

Apply the distributive property.

$\frac{a^{2} + 3 a - 10 - 3 a \cdot 2 - 2 \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.7

Multiply $2$ by $- 3$ .

$\frac{a^{2} + 3 a - 10 - 6 a - 2 \cdot 2}{2 (2 a + 1) (a + 5)}$

Step 6.8

Multiply $- 2$ by $2$ .

$\frac{a^{2} + 3 a - 10 - 6 a - 4}{2 (2 a + 1) (a + 5)}$

Step 6.9

Subtract $6 a$ from $3 a$ .

$\frac{a^{2} - 3 a - 10 - 4}{2 (2 a + 1) (a + 5)}$

Step 6.10

Subtract $4$ from $- 10$ .

$\frac{a^{2} - 3 a - 14}{2 (2 a + 1) (a + 5)}$