Algebra Examples

$\frac{x}{3 x^{2} - 11 x - 42} - \frac{9 x}{5 x^{2} - 22 x - 48}$

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 $a \cdot c = 3 \cdot - 42 = - 126$ and whose sum is $b = - 11$ .

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Step 1.1.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{x}{3 x^{2} - 11 (x) - 42} - \frac{9 x}{5 x^{2} - 22 x - 48}$

Step 1.1.1.2

Rewrite $- 11$ as $7$ plus $- 18$

$\frac{x}{3 x^{2} + (7 - 18) x - 42} - \frac{9 x}{5 x^{2} - 22 x - 48}$

Step 1.1.1.3

Apply the distributive property.

$\frac{x}{3 x^{2} + 7 x - 18 x - 42} - \frac{9 x}{5 x^{2} - 22 x - 48}$

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.

$\frac{x}{(3 x^{2} + 7 x) - 18 x - 42} - \frac{9 x}{5 x^{2} - 22 x - 48}$

Step 1.1.2.2

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

$\frac{x}{x (3 x + 7) - 6 (3 x + 7)} - \frac{9 x}{5 x^{2} - 22 x - 48}$

Step 1.1.3

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

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{5 x^{2} - 22 x - 48}$

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 $a \cdot c = 5 \cdot - 48 = - 240$ and whose sum is $b = - 22$ .

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Step 1.2.1.1

Factor $- 22$ out of $- 22 x$ .

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{5 x^{2} - 22 (x) - 48}$

Step 1.2.1.2

Rewrite $- 22$ as $8$ plus $- 30$

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{5 x^{2} + (8 - 30) x - 48}$

Step 1.2.1.3

Apply the distributive property.

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{5 x^{2} + 8 x - 30 x - 48}$

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.

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{(5 x^{2} + 8 x) - 30 x - 48}$

Step 1.2.2.2

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

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{x (5 x + 8) - 6 (5 x + 8)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 8$ .

$\frac{x}{(3 x + 7) (x - 6)} - \frac{9 x}{(5 x + 8) (x - 6)}$

Step 2

To write $\frac{x}{(3 x + 7) (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{5 x + 8}{5 x + 8}$ .

$\frac{x}{(3 x + 7) (x - 6)} \cdot \frac{5 x + 8}{5 x + 8} - \frac{9 x}{(5 x + 8) (x - 6)}$

Step 3

To write $- \frac{9 x}{(5 x + 8) (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{3 x + 7}{3 x + 7}$ .

$\frac{x}{(3 x + 7) (x - 6)} \cdot \frac{5 x + 8}{5 x + 8} - \frac{9 x}{(5 x + 8) (x - 6)} \cdot \frac{3 x + 7}{3 x + 7}$

Step 4

Write each expression with a common denominator of $(3 x + 7) (x - 6) (5 x + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{(3 x + 7) (x - 6)}$ by $\frac{5 x + 8}{5 x + 8}$ .

$\frac{x (5 x + 8)}{(3 x + 7) (x - 6) (5 x + 8)} - \frac{9 x}{(5 x + 8) (x - 6)} \cdot \frac{3 x + 7}{3 x + 7}$

Step 4.2

Multiply $\frac{9 x}{(5 x + 8) (x - 6)}$ by $\frac{3 x + 7}{3 x + 7}$ .

$\frac{x (5 x + 8)}{(3 x + 7) (x - 6) (5 x + 8)} - \frac{9 x (3 x + 7)}{(5 x + 8) (x - 6) (3 x + 7)}$

Step 4.3

Reorder the factors of $(3 x + 7) (x - 6) (5 x + 8)$ .

$\frac{x (5 x + 8)}{(3 x + 7) (5 x + 8) (x - 6)} - \frac{9 x (3 x + 7)}{(5 x + 8) (x - 6) (3 x + 7)}$

Step 4.4

Reorder the factors of $(5 x + 8) (x - 6) (3 x + 7)$ .

$\frac{x (5 x + 8)}{(3 x + 7) (5 x + 8) (x - 6)} - \frac{9 x (3 x + 7)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (5 x + 8) - 9 x (3 x + 7)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $x (5 x + 8) - 9 x (3 x + 7)$ .

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Step 6.1.1

Factor $x$ out of $- 9 x (3 x + 7)$ .

$\frac{x (5 x + 8) + x (- 9 (3 x + 7))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.1.2

Factor $x$ out of $x (5 x + 8) + x (- 9 (3 x + 7))$ .

$\frac{x (5 x + 8 - 9 (3 x + 7))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.2

Apply the distributive property.

$\frac{x (5 x + 8 - 9 (3 x) - 9 \cdot 7)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.3

Multiply $3$ by $- 9$ .

$\frac{x (5 x + 8 - 27 x - 9 \cdot 7)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.4

Multiply $- 9$ by $7$ .

$\frac{x (5 x + 8 - 27 x - 63)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.5

Subtract $27 x$ from $5 x$ .

$\frac{x (- 22 x + 8 - 63)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.6

Subtract $63$ from $8$ .

$\frac{x (- 22 x - 55)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.7

Factor $11$ out of $- 22 x - 55$ .

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Step 6.7.1

Factor $11$ out of $- 22 x$ .

$\frac{x (11 (- 2 x) - 55)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.7.2

Factor $11$ out of $- 55$ .

$\frac{x (11 (- 2 x) + 11 (- 5))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 6.7.3

Factor $11$ out of $11 (- 2 x) + 11 (- 5)$ .

$\frac{x (11 (- 2 x - 5))}{(3 x + 7) (5 x + 8) (x - 6)}$

$\frac{x \cdot 11 (- 2 x - 5)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7

Simplify with factoring out.

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Step 7.1

Move $11$ to the left of $x$ .

$\frac{11 \cdot x (- 2 x - 5)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.2

Factor $- 1$ out of $- 2 x$ .

$\frac{11 x (- (2 x) - 5)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.3

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{11 x (- (2 x) - 1 (5))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.4

Factor $- 1$ out of $- (2 x) - 1 (5)$ .

$\frac{11 x (- (2 x + 5))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Rewrite $- (2 x + 5)$ as $- 1 (2 x + 5)$ .

$\frac{11 x (- 1 (2 x + 5))}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.5.2

Move the negative in front of the fraction.

$- \frac{(11 x) (2 x + 5)}{(3 x + 7) (5 x + 8) (x - 6)}$

Step 7.5.3

Reorder factors in $- \frac{(11 x) (2 x + 5)}{(3 x + 7) (5 x + 8) (x - 6)}$ .

$- \frac{11 x (2 x + 5)}{(3 x + 7) (5 x + 8) (x - 6)}$