Basic Math Examples

$\frac{y - 2}{y^{2} - 4 y + 4} \div \frac{y^{2} + 2 y}{y^{2} + 4 y + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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

$4 y = 2 \cdot y \cdot 2$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

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

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Step 3.1

Multiply by $1$ .

$\frac{(y - 2) \cdot 1}{{(y - 2)}^{2}} \cdot \frac{y^{2} + 4 y + 4}{y^{2} + 2 y}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\frac{(y - 2) \cdot 1}{(y - 2) (y - 2)} \cdot \frac{y^{2} + 4 y + 4}{y^{2} + 2 y}$

Step 3.2.2

Cancel the common factor.

$\frac{(y - 2) \cdot 1}{(y - 2) (y - 2)} \cdot \frac{y^{2} + 4 y + 4}{y^{2} + 2 y}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{y - 2} \cdot \frac{y^{2} + 4 y + 4}{y^{2} + 2 y}$

Step 4

Factor using the perfect square rule.

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Step 4.1

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

$\frac{1}{y - 2} \cdot \frac{y^{2} + 4 y + 2^{2}}{y^{2} + 2 y}$

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.

$4 y = 2 \cdot y \cdot 2$

Step 4.3

Rewrite the polynomial.

$\frac{1}{y - 2} \cdot \frac{y^{2} + 2 \cdot y \cdot 2 + 2^{2}}{y^{2} + 2 y}$

Step 4.4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Factor $y$ out of $2 y$ .

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

Step 5.1.3

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

$\frac{1}{y - 2} \cdot \frac{y + 2}{y}$

Step 5.3

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

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

Step 5.4

Reorder factors in $\frac{y + 2}{(y - 2) y}$ .

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