Algebra Examples

$\frac{6 y^{2} + 18 y - 60}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $6$ out of $6 y^{2} + 18 y - 60$ .

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Step 1.1.1

Factor $6$ out of $6 y^{2}$ .

$\frac{6 (y^{2}) + 18 y - 60}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1.1.2

Factor $6$ out of $18 y$ .

$\frac{6 (y^{2}) + 6 (3 y) - 60}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1.1.3

Factor $6$ out of $- 60$ .

$\frac{6 y^{2} + 6 (3 y) + 6 \cdot - 10}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1.1.4

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

$\frac{6 (y^{2} + 3 y) + 6 \cdot - 10}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1.1.5

Factor $6$ out of $6 (y^{2} + 3 y) + 6 \cdot - 10$ .

$\frac{6 (y^{2} + 3 y - 10)}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 1.2

Factor $y^{2} + 3 y - 10$ using the AC method.

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Step 1.2.1

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 $3$ .

$- 2, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{6 ((y - 2) (y + 5))}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

$\frac{6 (y - 2) (y + 5)}{3 y^{2} - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 2

Factor $3 y$ out of $3 y^{2} - 12 y$ .

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Step 2.1

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

$\frac{6 (y - 2) (y + 5)}{3 y (y) - 12 y} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 2.2

Factor $3 y$ out of $- 12 y$ .

$\frac{6 (y - 2) (y + 5)}{3 y (y) + 3 y (- 4)} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 2.3

Factor $3 y$ out of $3 y (y) + 3 y (- 4)$ .

$\frac{6 (y - 2) (y + 5)}{3 y (y - 4)} \cdot \frac{y^{2} - 16}{y^{2} + 2 y - 8}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{6 (y - 2) (y + 5)}{3 y (y - 4)} \cdot \frac{y^{2} - 4^{2}}{y^{2} + 2 y - 8}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 4$ .

$\frac{6 (y - 2) (y + 5)}{3 y (y - 4)} \cdot \frac{(y + 4) (y - 4)}{y^{2} + 2 y - 8}$

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 $2$ .

$- 2, 4$

Step 4.2

Write the factored form using these integers.

$\frac{6 (y - 2) (y + 5)}{3 y (y - 4)} \cdot \frac{(y + 4) (y - 4)}{(y - 2) (y + 4)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $y - 2$ .

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Step 5.1.1

Factor $y - 2$ out of $6 (y - 2) (y + 5)$ .

$\frac{(y - 2) (6 (y + 5))}{3 y (y - 4)} \cdot \frac{(y + 4) (y - 4)}{(y - 2) (y + 4)}$

Step 5.1.2

Cancel the common factor.

$\frac{(y - 2) (6 (y + 5))}{3 y (y - 4)} \cdot \frac{(y + 4) (y - 4)}{(y - 2) (y + 4)}$

Step 5.1.3

Rewrite the expression.

$\frac{6 (y + 5)}{3 y (y - 4)} \cdot \frac{(y + 4) (y - 4)}{y + 4}$

Step 5.2

Cancel the common factor of $y - 4$ .

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Step 5.2.1

Factor $y - 4$ out of $3 y (y - 4)$ .

$\frac{6 (y + 5)}{(y - 4) (3 y)} \cdot \frac{(y + 4) (y - 4)}{y + 4}$

Step 5.2.2

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

$\frac{6 (y + 5)}{(y - 4) (3 y)} \cdot \frac{(y - 4) (y + 4)}{y + 4}$

Step 5.2.3

Cancel the common factor.

$\frac{6 (y + 5)}{(y - 4) (3 y)} \cdot \frac{(y - 4) (y + 4)}{y + 4}$

Step 5.2.4

Rewrite the expression.

$\frac{6 (y + 5)}{3 y} \cdot \frac{y + 4}{y + 4}$

Step 5.3

Multiply $\frac{6 (y + 5)}{3 y}$ by $\frac{y + 4}{y + 4}$ .

$\frac{6 (y + 5) (y + 4)}{3 y (y + 4)}$

Step 5.4

Cancel the common factor of $6$ and $3$ .

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Step 5.4.1

Factor $3$ out of $6 (y + 5) (y + 4)$ .

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

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $3$ out of $3 y (y + 4)$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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

Step 5.5

Cancel the common factor of $y + 4$ .

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Step 5.5.1

Cancel the common factor.

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

Step 5.5.2

Rewrite the expression.

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