Pre-Algebra Examples

$\frac{y^{2} - 18 y + 80}{y^{2} - 10 y} \cdot \frac{y}{8 - y}$

Step 1

Factor $y^{2} - 18 y + 80$ using the AC method.

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Step 1.1

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 $80$ and whose sum is $- 18$ .

$- 10, - 8$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 10) (y - 8)}{y^{2} - 10 y} \cdot \frac{y}{8 - y}$

Step 2

Simplify terms.

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Step 2.1

Factor $y$ out of $y^{2} - 10 y$ .

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Step 2.1.1

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

$\frac{(y - 10) (y - 8)}{y \cdot y - 10 y} \cdot \frac{y}{8 - y}$

Step 2.1.2

Factor $y$ out of $- 10 y$ .

$\frac{(y - 10) (y - 8)}{y \cdot y + y \cdot - 10} \cdot \frac{y}{8 - y}$

Step 2.1.3

Factor $y$ out of $y \cdot y + y \cdot - 10$ .

$\frac{(y - 10) (y - 8)}{y (y - 10)} \cdot \frac{y}{8 - y}$

Step 2.2

Cancel the common factor of $y$ .

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Step 2.2.1

Cancel the common factor.

$\frac{(y - 10) (y - 8)}{y (y - 10)} \cdot \frac{y}{8 - y}$

Step 2.2.2

Rewrite the expression.

$\frac{(y - 10) (y - 8)}{y - 10} \cdot \frac{1}{8 - y}$

Step 2.3

Multiply $\frac{(y - 10) (y - 8)}{y - 10}$ by $\frac{1}{8 - y}$ .

$\frac{(y - 10) (y - 8)}{(y - 10) (8 - y)}$

Step 2.4

Cancel the common factor of $y - 10$ .

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Step 2.4.1

Cancel the common factor.

$\frac{(y - 10) (y - 8)}{(y - 10) (8 - y)}$

Step 2.4.2

Rewrite the expression.

$\frac{y - 8}{8 - y}$

Step 2.5

Cancel the common factor of $y - 8$ and $8 - y$ .

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Step 2.5.1

Factor $- 1$ out of $y$ .

$\frac{- 1 (- y) - 8}{8 - y}$

Step 2.5.2

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- 1 (- y) - 1 (8)}{8 - y}$

Step 2.5.3

Factor $- 1$ out of $- 1 (- y) - 1 (8)$ .

$\frac{- 1 (- y + 8)}{8 - y}$

Step 2.5.4

Reorder terms.

$\frac{- 1 (- y + 8)}{- y + 8}$

Step 2.5.5

Cancel the common factor.

$\frac{- 1 (- y + 8)}{- y + 8}$

Step 2.5.6

Divide $- 1$ by $1$ .

$- 1$