Finite Math Examples

$\frac{x}{3 x^{2} + 4 x - 32} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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 $a \cdot c = 3 \cdot - 32 = - 96$ and whose sum is $b = 4$ .

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Step 1.1.1.1

Factor $4$ out of $4 x$ .

$\frac{x}{3 x^{2} + 4 (x) - 32} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.1.1.2

Rewrite $4$ as $- 8$ plus $12$

$\frac{x}{3 x^{2} + (- 8 + 12) x - 32} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.1.1.3

Apply the distributive property.

$\frac{x}{3 x^{2} - 8 x + 12 x - 32} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{x}{(3 x^{2} - 8 x) + 12 x - 32} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.1.2.2

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

$\frac{x}{x (3 x - 8) + 4 (3 x - 8)} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 8$ .

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{3 x^{2} + 5 x - 28}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = 3 \cdot - 28 = - 84$ and whose sum is $b = 5$ .

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Step 1.2.1.1

Factor $5$ out of $5 x$ .

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{3 x^{2} + 5 (x) - 28}$

Step 1.2.1.2

Rewrite $5$ as $- 7$ plus $12$

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{3 x^{2} + (- 7 + 12) x - 28}$

Step 1.2.1.3

Apply the distributive property.

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{3 x^{2} - 7 x + 12 x - 28}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{(3 x^{2} - 7 x) + 12 x - 28}$

Step 1.2.2.2

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

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{x (3 x - 7) + 4 (3 x - 7)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 7$ .

$\frac{x}{(3 x - 8) (x + 4)} - \frac{8 x}{(3 x - 7) (x + 4)}$

Step 2

To write $\frac{x}{(3 x - 8) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 7}{3 x - 7}$ .

$\frac{x}{(3 x - 8) (x + 4)} \cdot \frac{3 x - 7}{3 x - 7} - \frac{8 x}{(3 x - 7) (x + 4)}$

Step 3

To write $- \frac{8 x}{(3 x - 7) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 8}{3 x - 8}$ .

$\frac{x}{(3 x - 8) (x + 4)} \cdot \frac{3 x - 7}{3 x - 7} - \frac{8 x}{(3 x - 7) (x + 4)} \cdot \frac{3 x - 8}{3 x - 8}$

Step 4

Write each expression with a common denominator of $(3 x - 8) (x + 4) (3 x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{(3 x - 8) (x + 4)}$ by $\frac{3 x - 7}{3 x - 7}$ .

$\frac{x (3 x - 7)}{(3 x - 8) (x + 4) (3 x - 7)} - \frac{8 x}{(3 x - 7) (x + 4)} \cdot \frac{3 x - 8}{3 x - 8}$

Step 4.2

Multiply $\frac{8 x}{(3 x - 7) (x + 4)}$ by $\frac{3 x - 8}{3 x - 8}$ .

$\frac{x (3 x - 7)}{(3 x - 8) (x + 4) (3 x - 7)} - \frac{8 x (3 x - 8)}{(3 x - 7) (x + 4) (3 x - 8)}$

Step 4.3

Reorder the factors of $(3 x - 8) (x + 4) (3 x - 7)$ .

$\frac{x (3 x - 7)}{(3 x - 7) (3 x - 8) (x + 4)} - \frac{8 x (3 x - 8)}{(3 x - 7) (x + 4) (3 x - 8)}$

Step 4.4

Reorder the factors of $(3 x - 7) (x + 4) (3 x - 8)$ .

$\frac{x (3 x - 7)}{(3 x - 7) (3 x - 8) (x + 4)} - \frac{8 x (3 x - 8)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (3 x - 7) - 8 x (3 x - 8)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $x (3 x - 7) - 8 x (3 x - 8)$ .

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Step 6.1.1

Factor $x$ out of $- 8 x (3 x - 8)$ .

$\frac{x (3 x - 7) + x (- 8 (3 x - 8))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.1.2

Factor $x$ out of $x (3 x - 7) + x (- 8 (3 x - 8))$ .

$\frac{x (3 x - 7 - 8 (3 x - 8))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.2

Apply the distributive property.

$\frac{x (3 x - 7 - 8 (3 x) - 8 \cdot - 8)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.3

Multiply $3$ by $- 8$ .

$\frac{x (3 x - 7 - 24 x - 8 \cdot - 8)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.4

Multiply $- 8$ by $- 8$ .

$\frac{x (3 x - 7 - 24 x + 64)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.5

Subtract $24 x$ from $3 x$ .

$\frac{x (- 21 x - 7 + 64)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.6

Add $- 7$ and $64$ .

$\frac{x (- 21 x + 57)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.7

Factor $3$ out of $- 21 x + 57$ .

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Step 6.7.1

Factor $3$ out of $- 21 x$ .

$\frac{x (3 (- 7 x) + 57)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.7.2

Factor $3$ out of $57$ .

$\frac{x (3 (- 7 x) + 3 (19))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 6.7.3

Factor $3$ out of $3 (- 7 x) + 3 (19)$ .

$\frac{x (3 (- 7 x + 19))}{(3 x - 7) (3 x - 8) (x + 4)}$

$\frac{x \cdot 3 (- 7 x + 19)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7

Simplify with factoring out.

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Step 7.1

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

$\frac{3 \cdot x (- 7 x + 19)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.2

Factor $- 1$ out of $- 7 x$ .

$\frac{3 x (- (7 x) + 19)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.3

Rewrite $19$ as $- 1 (- 19)$ .

$\frac{3 x (- (7 x) - 1 (- 19))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.4

Factor $- 1$ out of $- (7 x) - 1 (- 19)$ .

$\frac{3 x (- (7 x - 19))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Rewrite $- (7 x - 19)$ as $- 1 (7 x - 19)$ .

$\frac{3 x (- 1 (7 x - 19))}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.5.2

Move the negative in front of the fraction.

$- \frac{(3 x) (7 x - 19)}{(3 x - 7) (3 x - 8) (x + 4)}$

Step 7.5.3

Reorder factors in $- \frac{(3 x) (7 x - 19)}{(3 x - 7) (3 x - 8) (x + 4)}$ .

$- \frac{3 x (7 x - 19)}{(3 x - 7) (3 x - 8) (x + 4)}$