Precalculus Examples

$\frac{6 x}{7 x^{2} - 6 x - 16} - \frac{5 x}{7 x^{2} - 15 x + 2}$

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 $a \cdot c = 7 \cdot - 16 = - 112$ and whose sum is $b = - 6$ .

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Step 1.1.1.1

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

$\frac{6 x}{7 x^{2} - 6 (x) - 16} - \frac{5 x}{7 x^{2} - 15 x + 2}$

Step 1.1.1.2

Rewrite $- 6$ as $8$ plus $- 14$

$\frac{6 x}{7 x^{2} + (8 - 14) x - 16} - \frac{5 x}{7 x^{2} - 15 x + 2}$

Step 1.1.1.3

Apply the distributive property.

$\frac{6 x}{7 x^{2} + 8 x - 14 x - 16} - \frac{5 x}{7 x^{2} - 15 x + 2}$

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.

$\frac{6 x}{(7 x^{2} + 8 x) - 14 x - 16} - \frac{5 x}{7 x^{2} - 15 x + 2}$

Step 1.1.2.2

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

$\frac{6 x}{x (7 x + 8) - 2 (7 x + 8)} - \frac{5 x}{7 x^{2} - 15 x + 2}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 8$ .

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{7 x^{2} - 15 x + 2}$

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 $a \cdot c = 7 \cdot 2 = 14$ and whose sum is $b = - 15$ .

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Step 1.2.1.1

Factor $- 15$ out of $- 15 x$ .

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{7 x^{2} - 15 (x) + 2}$

Step 1.2.1.2

Rewrite $- 15$ as $- 1$ plus $- 14$

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{7 x^{2} + (- 1 - 14) x + 2}$

Step 1.2.1.3

Apply the distributive property.

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{7 x^{2} - 1 x - 14 x + 2}$

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.

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{(7 x^{2} - 1 x) - 14 x + 2}$

Step 1.2.2.2

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

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{x (7 x - 1) - 2 (7 x - 1)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 1$ .

$\frac{6 x}{(7 x + 8) (x - 2)} - \frac{5 x}{(7 x - 1) (x - 2)}$

Step 2

To write $\frac{6 x}{(7 x + 8) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{7 x - 1}{7 x - 1}$ .

$\frac{6 x}{(7 x + 8) (x - 2)} \cdot \frac{7 x - 1}{7 x - 1} - \frac{5 x}{(7 x - 1) (x - 2)}$

Step 3

To write $- \frac{5 x}{(7 x - 1) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{7 x + 8}{7 x + 8}$ .

$\frac{6 x}{(7 x + 8) (x - 2)} \cdot \frac{7 x - 1}{7 x - 1} - \frac{5 x}{(7 x - 1) (x - 2)} \cdot \frac{7 x + 8}{7 x + 8}$

Step 4

Write each expression with a common denominator of $(7 x + 8) (x - 2) (7 x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6 x}{(7 x + 8) (x - 2)}$ by $\frac{7 x - 1}{7 x - 1}$ .

$\frac{6 x (7 x - 1)}{(7 x + 8) (x - 2) (7 x - 1)} - \frac{5 x}{(7 x - 1) (x - 2)} \cdot \frac{7 x + 8}{7 x + 8}$

Step 4.2

Multiply $\frac{5 x}{(7 x - 1) (x - 2)}$ by $\frac{7 x + 8}{7 x + 8}$ .

$\frac{6 x (7 x - 1)}{(7 x + 8) (x - 2) (7 x - 1)} - \frac{5 x (7 x + 8)}{(7 x - 1) (x - 2) (7 x + 8)}$

Step 4.3

Reorder the factors of $(7 x + 8) (x - 2) (7 x - 1)$ .

$\frac{6 x (7 x - 1)}{(7 x + 8) (7 x - 1) (x - 2)} - \frac{5 x (7 x + 8)}{(7 x - 1) (x - 2) (7 x + 8)}$

Step 4.4

Reorder the factors of $(7 x - 1) (x - 2) (7 x + 8)$ .

$\frac{6 x (7 x - 1)}{(7 x + 8) (7 x - 1) (x - 2)} - \frac{5 x (7 x + 8)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 x (7 x - 1) - 5 x (7 x + 8)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $6 x (7 x - 1) - 5 x (7 x + 8)$ .

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Step 6.1.1

Factor $x$ out of $6 x (7 x - 1)$ .

$\frac{x (6 (7 x - 1)) - 5 x (7 x + 8)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.1.2

Factor $x$ out of $- 5 x (7 x + 8)$ .

$\frac{x (6 (7 x - 1)) + x (- 5 (7 x + 8))}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.1.3

Factor $x$ out of $x (6 (7 x - 1)) + x (- 5 (7 x + 8))$ .

$\frac{x (6 (7 x - 1) - 5 (7 x + 8))}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.2

Apply the distributive property.

$\frac{x (6 (7 x) + 6 \cdot - 1 - 5 (7 x + 8))}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.3

Multiply $7$ by $6$ .

$\frac{x (42 x + 6 \cdot - 1 - 5 (7 x + 8))}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.4

Multiply $6$ by $- 1$ .

$\frac{x (42 x - 6 - 5 (7 x + 8))}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.5

Apply the distributive property.

$\frac{x (42 x - 6 - 5 (7 x) - 5 \cdot 8)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.6

Multiply $7$ by $- 5$ .

$\frac{x (42 x - 6 - 35 x - 5 \cdot 8)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.7

Multiply $- 5$ by $8$ .

$\frac{x (42 x - 6 - 35 x - 40)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.8

Subtract $35 x$ from $42 x$ .

$\frac{x (7 x - 6 - 40)}{(7 x + 8) (7 x - 1) (x - 2)}$

Step 6.9

Subtract $40$ from $- 6$ .

$\frac{x (7 x - 46)}{(7 x + 8) (7 x - 1) (x - 2)}$