Finite Math Examples

$\frac{4 w^{2} - 20 w + 24}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4$ out of $4 w^{2} - 20 w + 24$ .

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Step 1.1.1

Factor $4$ out of $4 w^{2}$ .

$\frac{4 (w^{2}) - 20 w + 24}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1.1.2

Factor $4$ out of $- 20 w$ .

$\frac{4 (w^{2}) + 4 (- 5 w) + 24}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1.1.3

Factor $4$ out of $24$ .

$\frac{4 w^{2} + 4 (- 5 w) + 4 \cdot 6}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1.1.4

Factor $4$ out of $4 w^{2} + 4 (- 5 w)$ .

$\frac{4 (w^{2} - 5 w) + 4 \cdot 6}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1.1.5

Factor $4$ out of $4 (w^{2} - 5 w) + 4 \cdot 6$ .

$\frac{4 (w^{2} - 5 w + 6)}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 1.2

Factor $w^{2} - 5 w + 6$ using the AC method.

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Step 1.2.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2.2

Write the factored form using these integers.

$\frac{4 ((w - 3) (w - 2))}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 w^{2} - 48 w + 90}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $6$ out of $6 w^{2} - 48 w + 90$ .

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Step 2.1.1

Factor $6$ out of $6 w^{2}$ .

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 (w^{2}) - 48 w + 90}$

Step 2.1.2

Factor $6$ out of $- 48 w$ .

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 (w^{2}) + 6 (- 8 w) + 90}$

Step 2.1.3

Factor $6$ out of $90$ .

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 w^{2} + 6 (- 8 w) + 6 \cdot 15}$

Step 2.1.4

Factor $6$ out of $6 w^{2} + 6 (- 8 w)$ .

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 (w^{2} - 8 w) + 6 \cdot 15}$

Step 2.1.5

Factor $6$ out of $6 (w^{2} - 8 w) + 6 \cdot 15$ .

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 (w^{2} - 8 w + 15)}$

Step 2.2

Factor $w^{2} - 8 w + 15$ using the AC method.

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Step 2.2.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.2.2

Write the factored form using these integers.

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 ((w - 5) (w - 3))}$

$\frac{4 (w - 3) (w - 2)}{w^{2}} \cdot \frac{w}{6 (w - 5) (w - 3)}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{4 (w - 3) (w - 2) w}{w^{2} (6 (w - 5) (w - 3))}$

Step 3.2

Cancel the common factor of $4$ and $6$ .

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Step 3.2.1

Factor $2$ out of $4 (w - 3) (w - 2) w$ .

$\frac{2 (2 (w - 3) (w - 2) w)}{w^{2} (6 (w - 5) (w - 3))}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $w^{2} (6 (w - 5) (w - 3))$ .

$\frac{2 (2 (w - 3) (w - 2) w)}{2 (w^{2} (3 (w - 5) (w - 3)))}$

Step 3.2.2.2

Cancel the common factor.

$\frac{2 (2 (w - 3) (w - 2) w)}{2 (w^{2} (3 (w - 5) (w - 3)))}$

Step 3.2.2.3

Rewrite the expression.

$\frac{2 (w - 3) (w - 2) w}{w^{2} (3 (w - 5) (w - 3))}$

Step 3.3

Cancel the common factor of $w - 3$ .

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Step 3.3.1

Cancel the common factor.

$\frac{2 (w - 3) (w - 2) w}{w^{2} (3 (w - 5) (w - 3))}$

Step 3.3.2

Rewrite the expression.

$\frac{(2 (w - 2)) w}{w^{2} (3 (w - 5))}$

Step 3.4

Cancel the common factor of $w$ and $w^{2}$ .

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Step 3.4.1

Factor $w$ out of $(2 (w - 2)) w$ .

$\frac{w (2 (w - 2))}{w^{2} (3 (w - 5))}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $w$ out of $w^{2} (3 (w - 5))$ .

$\frac{w (2 (w - 2))}{w (w (3 (w - 5)))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{w (2 (w - 2))}{w (w (3 (w - 5)))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{2 (w - 2)}{w (3 (w - 5))}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite.

$\frac{2 (w - 2)}{w^{2} (6 (w - 5) (w - 3))}$

Step 4.2

Factor $2$ out of $6$ .

$\frac{2 (w - 2)}{w^{2} (2 (3) (w - 5) (w - 3))}$

Step 4.3

Rewrite.

$\frac{2 (w - 2)}{w^{2} (\frac{6}{2} (w - 5) (w - 3))}$

Step 4.4

Divide $6$ by $2$ .

$\frac{2 (w - 2)}{w^{2} (3 (w - 5) (w - 3))}$

Step 4.5

Rewrite.

$\frac{2 (w - 2)}{w^{2} (6 (w - 5))}$

Step 4.6

Factor $2$ out of $6$ .

$\frac{2 (w - 2)}{w^{2} (2 (3) (w - 5))}$

Step 4.7

Rewrite.

$\frac{2 (w - 2)}{w^{2} (\frac{6}{2} (w - 5))}$

Step 4.8

Divide $6$ by $2$ .

$\frac{2 (w - 2)}{w^{2} (3 (w - 5))}$

Step 4.9

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

$\frac{2 (w - 2)}{w \cdot w (3 (w - 5))}$

Step 4.10

Rewrite.

$\frac{2 (w - 2)}{w^{1} (3 (w - 5))}$

Step 4.11

Simplify.

$\frac{2 (w - 2)}{w (3 (w - 5))}$

Step 4.12

Remove unnecessary parentheses.

$\frac{2 (w - 2)}{w \cdot 3 (w - 5)}$

Step 5

Move $3$ to the left of $w$ .

$\frac{2 (w - 2)}{3 w (w - 5)}$