Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Factor $y$ out of $y^{1}$ .

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

Step 2.1.4

Factor $y$ out of $y \cdot y + y \cdot 1$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Cancel the common factor of $y$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

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 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Factor $4$ out of $4 y$ .

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

Step 4.1.2

Factor $4$ out of $4$ .

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

Step 4.1.3

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

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

Step 4.2

Cancel the common factor of $y + 1$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $y - 2$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

$\frac{y - 2}{4}$