Finite Math Examples

$\frac{\frac{y^{2} - y - 12}{y + 2}}{\frac{y - 4}{y^{2} - 4 y - 12}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - y - 12}{y + 2} \cdot \frac{y^{2} - 4 y - 12}{y - 4}$

Step 2

Factor $y^{2} - y - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 4) (y + 3)}{y + 2} \cdot \frac{y^{2} - 4 y - 12}{y - 4}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y - 4$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(y - 4) (y + 3)}{y + 2} \cdot \frac{y^{2} - 4 y - 12}{y - 4}$

Step 3.1.2

Rewrite the expression.

$\frac{y + 3}{y + 2} (y^{2} - 4 y - 12)$

Step 3.2

Multiply $\frac{y + 3}{y + 2}$ by $y^{2} - 4 y - 12$ .

$\frac{(y + 3) (y^{2} - 4 y - 12)}{y + 2}$

Step 4

Factor $y^{2} - 4 y - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(y + 3) ((y - 6) (y + 2))}{y + 2}$

$\frac{(y + 3) (y - 6) (y + 2)}{y + 2}$

Step 5

Cancel the common factor of $y + 2$ .

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Step 5.1

Cancel the common factor.

$\frac{(y + 3) (y - 6) (y + 2)}{y + 2}$

Step 5.2

Divide $(y + 3) (y - 6)$ by $1$ .

$(y + 3) (y - 6)$

Step 6

Expand $(y + 3) (y - 6)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$y (y - 6) + 3 (y - 6)$

Step 6.2

Apply the distributive property.

$y \cdot y + y \cdot - 6 + 3 (y - 6)$

Step 6.3

Apply the distributive property.

$y \cdot y + y \cdot - 6 + 3 y + 3 \cdot - 6$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 6 + 3 y + 3 \cdot - 6$

Step 7.1.2

Move $- 6$ to the left of $y$ .

$y^{2} - 6 \cdot y + 3 y + 3 \cdot - 6$

Step 7.1.3

Multiply $3$ by $- 6$ .

$y^{2} - 6 y + 3 y - 18$

Step 7.2

Add $- 6 y$ and $3 y$ .

$y^{2} - 3 y - 18$