Calculus Examples

$\frac{6 y^{2} + 13 y + 6}{20 - 5 y} \div \frac{4 y^{2} - 9}{y^{2} - 2 y - 8}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 y^{2} + 13 y + 6}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 6 \cdot 6 = 36$ and whose sum is $b = 13$ .

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Step 2.1.1

Factor $13$ out of $13 y$ .

$\frac{6 y^{2} + 13 (y) + 6}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2.1.2

Rewrite $13$ as $4$ plus $9$

$\frac{6 y^{2} + (4 + 9) y + 6}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2.1.3

Apply the distributive property.

$\frac{6 y^{2} + 4 y + 9 y + 6}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(6 y^{2} + 4 y) + 9 y + 6}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2.2.2

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

$\frac{2 y (3 y + 2) + 3 (3 y + 2)}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 y + 2$ .

$\frac{(3 y + 2) (2 y + 3)}{20 - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 3

Factor $5$ out of $20 - 5 y$ .

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Step 3.1

Factor $5$ out of $20$ .

$\frac{(3 y + 2) (2 y + 3)}{5 (4) - 5 y} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 3.2

Factor $5$ out of $- 5 y$ .

$\frac{(3 y + 2) (2 y + 3)}{5 (4) + 5 (- y)} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 3.3

Factor $5$ out of $5 (4) + 5 (- y)$ .

$\frac{(3 y + 2) (2 y + 3)}{5 (4 - y)} \cdot \frac{y^{2} - 2 y - 8}{4 y^{2} - 9}$

Step 4

Factor $y^{2} - 2 y - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

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

Step 5.2

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

$\frac{(3 y + 2) (2 y + 3)}{5 (4 - y)} \cdot \frac{(y - 4) (y + 2)}{{(2 y)}^{2} - 3^{2}}$

Step 5.3

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

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

Step 6

Cancel the common factor of $2 y + 3$ .

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Step 6.1

Factor $2 y + 3$ out of $(3 y + 2) (2 y + 3)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\frac{3 y + 2}{5 (4 - y)}$ by $\frac{(y - 4) (y + 2)}{2 y - 3}$ .

$\frac{(3 y + 2) ((y - 4) (y + 2))}{5 (4 - y) (2 y - 3)}$

Step 8

Cancel the common factor of $y - 4$ and $4 - y$ .

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Step 8.1

Factor $- 1$ out of $y$ .

$\frac{(3 y + 2) ((- 1 (- y) - 4) (y + 2))}{5 (4 - y) (2 y - 3)}$

Step 8.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(3 y + 2) ((- 1 (- y) - 1 (4)) (y + 2))}{5 (4 - y) (2 y - 3)}$

Step 8.3

Factor $- 1$ out of $- 1 (- y) - 1 (4)$ .

$\frac{(3 y + 2) (- 1 (- y + 4) (y + 2))}{5 (4 - y) (2 y - 3)}$

Step 8.4

Reorder terms.

$\frac{(3 y + 2) (- 1 (- y + 4) (y + 2))}{5 (- y + 4) (2 y - 3)}$

Step 8.5

Cancel the common factor.

$\frac{(3 y + 2) (- 1 (- y + 4) (y + 2))}{5 (- y + 4) (2 y - 3)}$

Step 8.6

Rewrite the expression.

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

Step 9

Simplify the numerator.

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Step 9.1

Rewrite.

$\frac{(3 y + 2) ((y - 4) (y + 2))}{5 (2 y - 3)}$

Step 9.2

Factor $- 1$ out of $y$ .

$\frac{(3 y + 2) ((- 1 (- y) - 4) (y + 2))}{5 (2 y - 3)}$

Step 9.3

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(3 y + 2) ((- 1 (- y) - 1 (4)) (y + 2))}{5 (2 y - 3)}$

Step 9.4

Factor $- 1$ out of $- 1 (- y) - 1 (4)$ .

$\frac{(3 y + 2) (- 1 (- y + 4) (y + 2))}{5 (2 y - 3)}$

Step 9.5

Rewrite.

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

Step 9.6

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

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

Step 9.7

Remove unnecessary parentheses.

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

Step 9.8

Factor out negative.

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

Step 10

Move the negative in front of the fraction.

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