Basic Math Examples

$\frac{y^{2} - 9}{4 y + 12} \div \frac{2 y^{2} - 12 y + 18}{8 y + 24}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 9}{4 y + 12} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{y^{2} - 3^{2}}{4 y + 12} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 2.2

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 $b = 3$ .

$\frac{(y + 3) (y - 3)}{4 y + 12} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3

Simplify terms.

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Step 3.1

Factor $4$ out of $4 y + 12$ .

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Step 3.1.1

Factor $4$ out of $4 y$ .

$\frac{(y + 3) (y - 3)}{4 (y) + 12} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3.1.2

Factor $4$ out of $12$ .

$\frac{(y + 3) (y - 3)}{4 y + 4 \cdot 3} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3.1.3

Factor $4$ out of $4 y + 4 \cdot 3$ .

$\frac{(y + 3) (y - 3)}{4 (y + 3)} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3.2

Cancel the common factor of $y + 3$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(y + 3) (y - 3)}{4 (y + 3)} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3.2.2

Rewrite the expression.

$\frac{y - 3}{4} \cdot \frac{8 y + 24}{2 y^{2} - 12 y + 18}$

Step 3.3

Cancel the common factor of $8 y + 24$ and $2 y^{2} - 12 y + 18$ .

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Step 3.3.1

Factor $2$ out of $8 y$ .

$\frac{y - 3}{4} \cdot \frac{2 (4 y) + 24}{2 y^{2} - 12 y + 18}$

Step 3.3.2

Factor $2$ out of $24$ .

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

Step 3.3.3

Factor $2$ out of $2 (4 y) + 2 (12)$ .

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

Step 3.3.4

Cancel the common factors.

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Step 3.3.4.1

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

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

Step 3.3.4.2

Factor $2$ out of $- 12 y$ .

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

Step 3.3.4.3

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

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

Step 3.3.4.4

Factor $2$ out of $18$ .

$\frac{y - 3}{4} \cdot \frac{2 (4 y + 12)}{2 (y^{2} - 6 y) + 2 \cdot 9}$

Step 3.3.4.5

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

$\frac{y - 3}{4} \cdot \frac{2 (4 y + 12)}{2 (y^{2} - 6 y + 9)}$

Step 3.3.4.6

Cancel the common factor.

$\frac{y - 3}{4} \cdot \frac{2 (4 y + 12)}{2 (y^{2} - 6 y + 9)}$

Step 3.3.4.7

Rewrite the expression.

$\frac{y - 3}{4} \cdot \frac{4 y + 12}{y^{2} - 6 y + 9}$

Step 3.4

Factor $4$ out of $4 y + 12$ .

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Step 3.4.1

Factor $4$ out of $4 y$ .

$\frac{y - 3}{4} \cdot \frac{4 (y) + 12}{y^{2} - 6 y + 9}$

Step 3.4.2

Factor $4$ out of $12$ .

$\frac{y - 3}{4} \cdot \frac{4 y + 4 \cdot 3}{y^{2} - 6 y + 9}$

Step 3.4.3

Factor $4$ out of $4 y + 4 \cdot 3$ .

$\frac{y - 3}{4} \cdot \frac{4 (y + 3)}{y^{2} - 6 y + 9}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

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

Step 4.2

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 4.3

Rewrite the polynomial.

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

Step 4.4

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

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $y - 3$ .

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Step 5.1.1

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

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $4$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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