Pre-Algebra Examples

$\frac{y^{2} + 3 y + 2}{y^{2} + 11 y + 18} \div \frac{y^{2} - 12 y + 35}{y^{2} + 4 y - 45}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} + 3 y + 2}{y^{2} + 11 y + 18} \cdot \frac{y^{2} + 4 y - 45}{y^{2} - 12 y + 35}$

Step 2

Factor $y^{2} + 3 y + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(y + 1) (y + 2)}{y^{2} + 11 y + 18} \cdot \frac{y^{2} + 4 y - 45}{y^{2} - 12 y + 35}$

Step 3

Factor $y^{2} + 11 y + 18$ 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 $18$ and whose sum is $11$ .

$2, 9$

Step 3.2

Write the factored form using these integers.

$\frac{(y + 1) (y + 2)}{(y + 2) (y + 9)} \cdot \frac{y^{2} + 4 y - 45}{y^{2} - 12 y + 35}$

Step 4

Cancel the common factor of $y + 2$ .

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Step 4.1

Cancel the common factor.

$\frac{(y + 1) (y + 2)}{(y + 2) (y + 9)} \cdot \frac{y^{2} + 4 y - 45}{y^{2} - 12 y + 35}$

Step 4.2

Rewrite the expression.

$\frac{y + 1}{y + 9} \cdot \frac{y^{2} + 4 y - 45}{y^{2} - 12 y + 35}$

Step 5

Factor $y^{2} + 4 y - 45$ 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 $- 45$ and whose sum is $4$ .

$- 5, 9$

Step 5.2

Write the factored form using these integers.

$\frac{y + 1}{y + 9} \cdot \frac{(y - 5) (y + 9)}{y^{2} - 12 y + 35}$

Step 6

Factor $y^{2} - 12 y + 35$ using the AC method.

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Step 6.1

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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 6.2

Write the factored form using these integers.

$\frac{y + 1}{y + 9} \cdot \frac{(y - 5) (y + 9)}{(y - 7) (y - 5)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $y + 9$ .

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Step 7.1.1

Factor $y + 9$ out of $(y - 5) (y + 9)$ .

$\frac{y + 1}{y + 9} \cdot \frac{(y + 9) (y - 5)}{(y - 7) (y - 5)}$

Step 7.1.2

Cancel the common factor.

$\frac{y + 1}{y + 9} \cdot \frac{(y + 9) (y - 5)}{(y - 7) (y - 5)}$

Step 7.1.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $y - 5$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$(y + 1) \frac{1}{y - 7}$

Step 7.3

Multiply $y + 1$ by $\frac{1}{y - 7}$ .

$\frac{y + 1}{y - 7}$