Pre-Algebra Examples

$\frac{\frac{y^{2} - 2 y - 24}{y^{2} - 8 y + 16}}{\frac{y - 6}{y - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 2 y - 24}{y^{2} - 8 y + 16} \cdot \frac{y - 4}{y - 6}$

Step 2

Factor $y^{2} - 2 y - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 6) (y + 4)}{y^{2} - 8 y + 16} \cdot \frac{y - 4}{y - 6}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{(y - 6) (y + 4)}{y^{2} - 8 y + 4^{2}} \cdot \frac{y - 4}{y - 6}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 y = 2 \cdot y \cdot 4$

Step 3.3

Rewrite the polynomial.

$\frac{(y - 6) (y + 4)}{y^{2} - 2 \cdot y \cdot 4 + 4^{2}} \cdot \frac{y - 4}{y - 6}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 4$ .

$\frac{(y - 6) (y + 4)}{{(y - 4)}^{2}} \cdot \frac{y - 4}{y - 6}$

Step 4

Cancel the common factor of $y - 6$ .

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Step 4.1

Cancel the common factor.

$\frac{(y - 6) (y + 4)}{{(y - 4)}^{2}} \cdot \frac{y - 4}{y - 6}$

Step 4.2

Rewrite the expression.

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

Step 5

Cancel the common factor of $y - 4$ .

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Step 5.1

Factor $y - 4$ out of ${(y - 4)}^{2}$ .

$\frac{y + 4}{(y - 4) (y - 4)} (y - 4)$

Step 5.2

Cancel the common factor.

$\frac{y + 4}{(y - 4) (y - 4)} (y - 4)$

Step 5.3

Rewrite the expression.

$\frac{y + 4}{y - 4}$