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.

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