Pre-Algebra Examples

$\frac{\frac{y^{2} + 9 y + 20}{y^{2} + 12 y + 32}}{\frac{y^{2} - 13 y + 42}{y^{2} + 2 y - 48}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} + 9 y + 20}{y^{2} + 12 y + 32} \cdot \frac{y^{2} + 2 y - 48}{y^{2} - 13 y + 42}$

Step 2

Factor $y^{2} + 9 y + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(y + 4) (y + 5)}{y^{2} + 12 y + 32} \cdot \frac{y^{2} + 2 y - 48}{y^{2} - 13 y + 42}$

Step 3

Factor $y^{2} + 12 y + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(y + 4) (y + 5)}{(y + 4) (y + 8)} \cdot \frac{y^{2} + 2 y - 48}{y^{2} - 13 y + 42}$

Step 4

Cancel the common factor of $y + 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(y + 4) (y + 5)}{(y + 4) (y + 8)} \cdot \frac{y^{2} + 2 y - 48}{y^{2} - 13 y + 42}$

Step 4.2

Rewrite the expression.

$\frac{y + 5}{y + 8} \cdot \frac{y^{2} + 2 y - 48}{y^{2} - 13 y + 42}$

Step 5

Factor $y^{2} + 2 y - 48$ 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 $- 48$ and whose sum is $2$ .

$- 6, 8$

Step 5.2

Write the factored form using these integers.

$\frac{y + 5}{y + 8} \cdot \frac{(y - 6) (y + 8)}{y^{2} - 13 y + 42}$

Step 6

Factor $y^{2} - 13 y + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 6.2

Write the factored form using these integers.

$\frac{y + 5}{y + 8} \cdot \frac{(y - 6) (y + 8)}{(y - 7) (y - 6)}$

Step 7

Cancel the common factor of $y + 8$ .

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Step 7.1

Factor $y + 8$ out of $(y - 6) (y + 8)$ .

$\frac{y + 5}{y + 8} \cdot \frac{(y + 8) (y - 6)}{(y - 7) (y - 6)}$

Step 7.2

Cancel the common factor.

$\frac{y + 5}{y + 8} \cdot \frac{(y + 8) (y - 6)}{(y - 7) (y - 6)}$

Step 7.3

Rewrite the expression.

$(y + 5) \frac{y - 6}{(y - 7) (y - 6)}$

Step 8

Cancel the common factor of $y - 6$ .

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Step 8.1

Cancel the common factor.

$(y + 5) \frac{y - 6}{(y - 7) (y - 6)}$

Step 8.2

Rewrite the expression.

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

Step 9

Multiply $y + 5$ by $\frac{1}{y - 7}$ .

$\frac{y + 5}{y - 7}$