Pre-Algebra Examples

$\frac{y^{2} + 7 y + 10}{y^{2} + 5 y} \div \frac{y^{2} + y - 2}{2 y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} + 7 y + 10}{y^{2} + 5 y} \cdot \frac{2 y}{y^{2} + y - 2}$

Step 2

Factor $y^{2} + 7 y + 10$ using the AC method.

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Step 2.1

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 $10$ and whose sum is $7$ .

$2, 5$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $y$ out of $5 y$ .

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

Step 3.1.3

Factor $y$ out of $y \cdot y + y \cdot 5$ .

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Factor $y$ out of $2 y$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Multiply $\frac{(y + 2) (y + 5)}{y + 5}$ by $\frac{2}{y^{2} + y - 2}$ .

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

Step 3.4

Cancel the common factor of $y + 5$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 4

Factor $y^{2} + y - 2$ using the AC method.

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Step 4.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 4.2

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $y + 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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