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

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Step 2.1

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

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

Step 2.2

Factor $y$ out of $5 y$ .

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

Step 2.3

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

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

Step 3

Cancel the common factor of $y$ .

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Step 3.1

Factor $y$ out of $2 y$ .

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

Multiply $\frac{y^{2} + 7 y - 10}{y + 5}$ by $\frac{2}{y^{2} + y - 2}$ .

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

Step 5

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

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Step 5.1

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 5.2

Write the factored form using these integers.

$\frac{(y^{2} + 7 y - 10) \cdot 2}{(y + 5) ((y - 1) (y + 2))}$

$\frac{(y^{2} + 7 y - 10) \cdot 2}{(y + 5) (y - 1) (y + 2)}$

Step 6

Move $2$ to the left of $y^{2} + 7 y - 10$ .

$\frac{2 (y^{2} + 7 y - 10)}{(y + 5) (y - 1) (y + 2)}$

Step 7

Apply the distributive property.

$\frac{2 y^{2} + 2 (7 y) + 2 \cdot - 10}{(y + 5) (y - 1) (y + 2)}$

Step 8

Simplify.

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Step 8.1

Multiply $7$ by $2$ .

$\frac{2 y^{2} + 14 y + 2 \cdot - 10}{(y + 5) (y - 1) (y + 2)}$

Step 8.2

Multiply $2$ by $- 10$ .

$\frac{2 y^{2} + 14 y - 20}{(y + 5) (y - 1) (y + 2)}$

Step 9

Split the fraction $\frac{2 y^{2} + 14 y - 20}{(y + 5) (y - 1) (y + 2)}$ into two fractions.

$\frac{2 y^{2} + 14 y}{(y + 5) (y - 1) (y + 2)} + \frac{- 20}{(y + 5) (y - 1) (y + 2)}$

Step 10

Split the fraction $\frac{2 y^{2} + 14 y}{(y + 5) (y - 1) (y + 2)}$ into two fractions.

$\frac{2 y^{2}}{(y + 5) (y - 1) (y + 2)} + \frac{14 y}{(y + 5) (y - 1) (y + 2)} + \frac{- 20}{(y + 5) (y - 1) (y + 2)}$

Step 11

Move the negative in front of the fraction.

$\frac{2 y^{2}}{(y + 5) (y - 1) (y + 2)} + \frac{14 y}{(y + 5) (y - 1) (y + 2)} - \frac{20}{(y + 5) (y - 1) (y + 2)}$