Basic Math Examples

$\frac{y + 2}{5 y^{2}} \div \frac{y^{2} - 4 y - 5}{25 y^{2} - 5 y^{3}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y + 2}{5 y^{2}} \cdot \frac{25 y^{2} - 5 y^{3}}{y^{2} - 4 y - 5}$

Step 2

Factor $5 y^{2}$ out of $25 y^{2} - 5 y^{3}$ .

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Step 2.1

Factor $5 y^{2}$ out of $25 y^{2}$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Factor $y^{2} - 4 y - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2

Write the factored form using these integers.

$\frac{y + 2}{5 y^{2}} \cdot \frac{5 y^{2} (5 - y)}{(y - 5) (y + 1)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $5 y^{2}$ .

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Step 4.1.1

Cancel the common factor.

$\frac{y + 2}{5 y^{2}} \cdot \frac{5 y^{2} (5 - y)}{(y - 5) (y + 1)}$

Step 4.1.2

Rewrite the expression.

$(y + 2) \frac{5 - y}{(y - 5) (y + 1)}$

Step 4.2

Cancel the common factor of $5 - y$ and $y - 5$ .

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Step 4.2.1

Rewrite $5$ as $- 1 (- 5)$ .

$(y + 2) \frac{- 1 (- 5) - y}{(y - 5) (y + 1)}$

Step 4.2.2

Factor $- 1$ out of $- y$ .

$(y + 2) \frac{- 1 (- 5) - (y)}{(y - 5) (y + 1)}$

Step 4.2.3

Factor $- 1$ out of $- 1 (- 5) - (y)$ .

$(y + 2) \frac{- 1 (- 5 + y)}{(y - 5) (y + 1)}$

Step 4.2.4

Reorder terms.

$(y + 2) \frac{- 1 (y - 5)}{(y - 5) (y + 1)}$

Step 4.2.5

Cancel the common factor.

$(y + 2) \frac{- 1 (y - 5)}{(y - 5) (y + 1)}$

Step 4.2.6

Rewrite the expression.

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

Step 4.3

Move the negative in front of the fraction.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Rewrite using the commutative property of multiplication.

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

Step 5

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

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Combine $- 2$ and $\frac{1}{y + 1}$ .

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

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{1}{y + 1}$ and $y$ .

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

Step 6.2

Move the negative in front of the fraction.

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

Step 7

Simplify terms.

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Step 7.1

Combine the numerators over the common denominator.

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

Step 7.2

Factor $- 1$ out of $- y$ .

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

Step 7.3

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 7.4

Factor $- 1$ out of $- (y) - 1 (2)$ .

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

Step 7.5

Simplify the expression.

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Step 7.5.1

Rewrite $- (y + 2)$ as $- 1 (y + 2)$ .

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

Step 7.5.2

Move the negative in front of the fraction.

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