Algebra Examples

$\frac{2 y^{2} - 6 y - 20}{4 y + 12} \div \frac{y^{2} + 5 y + 6}{3 y^{2} + 18 y + 27}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 y^{2} - 6 y - 20}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2

Cancel the common factor of $2 y^{2} - 6 y - 20$ and $4 y + 12$ .

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Step 2.1

Factor $2$ out of $2 y^{2}$ .

$\frac{2 (y^{2}) - 6 y - 20}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.2

Factor $2$ out of $- 6 y$ .

$\frac{2 (y^{2}) + 2 (- 3 y) - 20}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.3

Factor $2$ out of $2 (y^{2}) + 2 (- 3 y)$ .

$\frac{2 (y^{2} - 3 y) - 20}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.4

Factor $2$ out of $- 20$ .

$\frac{2 (y^{2} - 3 y) + 2 \cdot - 10}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.5

Factor $2$ out of $2 (y^{2} - 3 y) + 2 \cdot - 10$ .

$\frac{2 (y^{2} - 3 y - 10)}{4 y + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $2$ out of $4 y$ .

$\frac{2 (y^{2} - 3 y - 10)}{2 (2 y) + 12} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.6.2

Factor $2$ out of $12$ .

$\frac{2 (y^{2} - 3 y - 10)}{2 (2 y) + 2 (6)} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.6.3

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

$\frac{2 (y^{2} - 3 y - 10)}{2 (2 y + 6)} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.6.4

Cancel the common factor.

$\frac{2 (y^{2} - 3 y - 10)}{2 (2 y + 6)} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 2.6.5

Rewrite the expression.

$\frac{y^{2} - 3 y - 10}{2 y + 6} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 3

Factor $y^{2} - 3 y - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 3.2

Write the factored form using these integers.

$\frac{(y - 5) (y + 2)}{2 y + 6} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 4

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

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Step 4.1

Factor $2$ out of $2 y$ .

$\frac{(y - 5) (y + 2)}{2 (y) + 6} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 4.2

Factor $2$ out of $6$ .

$\frac{(y - 5) (y + 2)}{2 y + 2 \cdot 3} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 4.3

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

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 y^{2} + 18 y + 27}{y^{2} + 5 y + 6}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $3$ out of $3 y^{2} + 18 y + 27$ .

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Step 5.1.1

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

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2}) + 18 y + 27}{y^{2} + 5 y + 6}$

Step 5.1.2

Factor $3$ out of $18 y$ .

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2}) + 3 (6 y) + 27}{y^{2} + 5 y + 6}$

Step 5.1.3

Factor $3$ out of $27$ .

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 y^{2} + 3 (6 y) + 3 \cdot 9}{y^{2} + 5 y + 6}$

Step 5.1.4

Factor $3$ out of $3 y^{2} + 3 (6 y)$ .

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2} + 6 y) + 3 \cdot 9}{y^{2} + 5 y + 6}$

Step 5.1.5

Factor $3$ out of $3 (y^{2} + 6 y) + 3 \cdot 9$ .

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2} + 6 y + 9)}{y^{2} + 5 y + 6}$

Step 5.2

Factor using the perfect square rule.

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Step 5.2.1

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

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2} + 6 y + 3^{2})}{y^{2} + 5 y + 6}$

Step 5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 y = 2 \cdot y \cdot 3$

Step 5.2.3

Rewrite the polynomial.

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 (y^{2} + 2 \cdot y \cdot 3 + 3^{2})}{y^{2} + 5 y + 6}$

Step 5.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 3$ .

$\frac{(y - 5) (y + 2)}{2 (y + 3)} \cdot \frac{3 {(y + 3)}^{2}}{y^{2} + 5 y + 6}$

Step 6

Factor $y^{2} + 5 y + 6$ using the AC method.

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Step 6.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 6.2

Write the factored form using these integers.

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

Step 7

Combine.

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

Step 8

Cancel the common factor of $y + 2$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of ${(y + 3)}^{2}$ and $y + 3$ .

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Step 9.1

Factor $y + 3$ out of $(y - 5) (3 {(y + 3)}^{2})$ .

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

Step 9.2

Cancel the common factors.

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Step 9.2.1

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

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 10

Cancel the common factor of $y + 3$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Move $3$ to the left of $y - 5$ .

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

Step 12

Apply the distributive property.

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

Step 13

Multiply $3$ by $- 5$ .

$\frac{3 y - 15}{2}$

Step 14

Split the fraction $\frac{3 y - 15}{2}$ into two fractions.

$\frac{3 y}{2} + \frac{- 15}{2}$

Step 15

Move the negative in front of the fraction.

$\frac{3 y}{2} - \frac{15}{2}$