Finite Math Examples

$\frac{6 x - 18}{9 r^{2} + 6 r - 24} \cdot \frac{12 r}{4 r - 12}$

Step 1

Factor $6$ out of $6 x - 18$ .

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Step 1.1

Factor $6$ out of $6 x$ .

$\frac{6 (x) - 18}{9 r^{2} + 6 r - 24} \cdot \frac{12 r}{4 r - 12}$

Step 1.2

Factor $6$ out of $- 18$ .

$\frac{6 x + 6 \cdot - 3}{9 r^{2} + 6 r - 24} \cdot \frac{12 r}{4 r - 12}$

Step 1.3

Factor $6$ out of $6 x + 6 \cdot - 3$ .

$\frac{6 (x - 3)}{9 r^{2} + 6 r - 24} \cdot \frac{12 r}{4 r - 12}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $3$ out of $9 r^{2} + 6 r - 24$ .

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Step 2.1.1

Factor $3$ out of $9 r^{2}$ .

$\frac{6 (x - 3)}{3 (3 r^{2}) + 6 r - 24} \cdot \frac{12 r}{4 r - 12}$

Step 2.1.2

Factor $3$ out of $6 r$ .

$\frac{6 (x - 3)}{3 (3 r^{2}) + 3 (2 r) - 24} \cdot \frac{12 r}{4 r - 12}$

Step 2.1.3

Factor $3$ out of $- 24$ .

$\frac{6 (x - 3)}{3 (3 r^{2}) + 3 (2 r) + 3 (- 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.1.4

Factor $3$ out of $3 (3 r^{2}) + 3 (2 r)$ .

$\frac{6 (x - 3)}{3 (3 r^{2} + 2 r) + 3 (- 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.1.5

Factor $3$ out of $3 (3 r^{2} + 2 r) + 3 (- 8)$ .

$\frac{6 (x - 3)}{3 (3 r^{2} + 2 r - 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.2

Factor by grouping.

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Step 2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = 2$ .

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Step 2.2.1.1

Factor $2$ out of $2 r$ .

$\frac{6 (x - 3)}{3 (3 r^{2} + 2 (r) - 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.2.1.2

Rewrite $2$ as $- 4$ plus $6$

$\frac{6 (x - 3)}{3 (3 r^{2} + (- 4 + 6) r - 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.2.1.3

Apply the distributive property.

$\frac{6 (x - 3)}{3 (3 r^{2} - 4 r + 6 r - 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{6 (x - 3)}{3 ((3 r^{2} - 4 r) + 6 r - 8)} \cdot \frac{12 r}{4 r - 12}$

Step 2.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{6 (x - 3)}{3 (r (3 r - 4) + 2 (3 r - 4))} \cdot \frac{12 r}{4 r - 12}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 r - 4$ .

$\frac{6 (x - 3)}{3 ((3 r - 4) (r + 2))} \cdot \frac{12 r}{4 r - 12}$

$\frac{6 (x - 3)}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{4 r - 12}$

Step 3

Simplify terms.

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Step 3.1

Factor $4$ out of $4 r - 12$ .

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Step 3.1.1

Factor $4$ out of $4 r$ .

$\frac{6 (x - 3)}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{4 (r) - 12}$

Step 3.1.2

Factor $4$ out of $- 12$ .

$\frac{6 (x - 3)}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{4 r + 4 \cdot - 3}$

Step 3.1.3

Factor $4$ out of $4 r + 4 \cdot - 3$ .

$\frac{6 (x - 3)}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{4 (r - 3)}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Factor $2$ out of $6 (x - 3)$ .

$\frac{2 (3 (x - 3))}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{4 (r - 3)}$

Step 3.2.2

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

$\frac{2 (3 (x - 3))}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{2 (2 (r - 3))}$

Step 3.2.3

Cancel the common factor.

$\frac{2 (3 (x - 3))}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{2 (2 (r - 3))}$

Step 3.2.4

Rewrite the expression.

$\frac{3 (x - 3)}{3 (3 r - 4) (r + 2)} \cdot \frac{12 r}{2 (r - 3)}$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Factor $3$ out of $3 (3 r - 4) (r + 2)$ .

$\frac{3 (x - 3)}{3 ((3 r - 4) (r + 2))} \cdot \frac{12 r}{2 (r - 3)}$

Step 3.3.2

Factor $3$ out of $12 r$ .

$\frac{3 (x - 3)}{3 ((3 r - 4) (r + 2))} \cdot \frac{3 (4 r)}{2 (r - 3)}$

Step 3.3.3

Cancel the common factor.

$\frac{3 (x - 3)}{3 ((3 r - 4) (r + 2))} \cdot \frac{3 (4 r)}{2 (r - 3)}$

Step 3.3.4

Rewrite the expression.

$\frac{3 (x - 3)}{(3 r - 4) (r + 2)} \cdot \frac{4 r}{2 (r - 3)}$

Step 3.4

Multiply $\frac{3 (x - 3)}{(3 r - 4) (r + 2)}$ by $\frac{4 r}{2 (r - 3)}$ .

$\frac{3 (x - 3) (4 r)}{(3 r - 4) (r + 2) (2 (r - 3))}$

Step 3.5

Multiply $4$ by $3$ .

$\frac{12 (x - 3) r}{(3 r - 4) (r + 2) (2 (r - 3))}$

Step 3.6

Cancel the common factor of $12$ and $2$ .

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Step 3.6.1

Factor $2$ out of $12 (x - 3) r$ .

$\frac{2 (6 (x - 3) r)}{(3 r - 4) (r + 2) (2 (r - 3))}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $2$ out of $(3 r - 4) (r + 2) (2 (r - 3))$ .

$\frac{2 (6 (x - 3) r)}{2 ((3 r - 4) (r + 2) (r - 3))}$

Step 3.6.2.2

Cancel the common factor.

$\frac{2 (6 (x - 3) r)}{2 ((3 r - 4) (r + 2) (r - 3))}$

Step 3.6.2.3

Rewrite the expression.

$\frac{6 (x - 3) r}{(3 r - 4) (r + 2) (r - 3)}$

Step 3.7

Reorder factors in $\frac{6 (x - 3) r}{(3 r - 4) (r + 2) (r - 3)}$ .

$\frac{6 r (x - 3)}{(3 r - 4) (r + 2) (r - 3)}$