Algebra Examples

$\frac{2 x}{5 x^{2} - 37 x + 14} - \frac{3 x}{5 x^{2} - 33 x - 14}$

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 $a \cdot c = 5 \cdot 14 = 70$ and whose sum is $b = - 37$ .

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Step 1.1.1.1

Factor $- 37$ out of $- 37 x$ .

$\frac{2 x}{5 x^{2} - 37 (x) + 14} - \frac{3 x}{5 x^{2} - 33 x - 14}$

Step 1.1.1.2

Rewrite $- 37$ as $- 2$ plus $- 35$

$\frac{2 x}{5 x^{2} + (- 2 - 35) x + 14} - \frac{3 x}{5 x^{2} - 33 x - 14}$

Step 1.1.1.3

Apply the distributive property.

$\frac{2 x}{5 x^{2} - 2 x - 35 x + 14} - \frac{3 x}{5 x^{2} - 33 x - 14}$

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.

$\frac{2 x}{(5 x^{2} - 2 x) - 35 x + 14} - \frac{3 x}{5 x^{2} - 33 x - 14}$

Step 1.1.2.2

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

$\frac{2 x}{x (5 x - 2) - 7 (5 x - 2)} - \frac{3 x}{5 x^{2} - 33 x - 14}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$\frac{2 x}{(5 x - 2) (x - 7)} - \frac{3 x}{5 x^{2} - 33 x - 14}$

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 $a \cdot c = 5 \cdot - 14 = - 70$ and whose sum is $b = - 33$ .

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Step 1.2.1.1

Factor $- 33$ out of $- 33 x$ .

$\frac{2 x}{(5 x - 2) (x - 7)} - \frac{3 x}{5 x^{2} - 33 (x) - 14}$

Step 1.2.1.2

Rewrite $- 33$ as $2$ plus $- 35$

$\frac{2 x}{(5 x - 2) (x - 7)} - \frac{3 x}{5 x^{2} + (2 - 35) x - 14}$

Step 1.2.1.3

Apply the distributive property.

$\frac{2 x}{(5 x - 2) (x - 7)} - \frac{3 x}{5 x^{2} + 2 x - 35 x - 14}$

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.

$\frac{2 x}{(5 x - 2) (x - 7)} - \frac{3 x}{(5 x^{2} + 2 x) - 35 x - 14}$

Step 1.2.2.2

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

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

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 2$ .

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

Step 2

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

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

Step 3

To write $- \frac{3 x}{(5 x + 2) (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{5 x - 2}{5 x - 2}$ .

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

Step 4

Write each expression with a common denominator of $(5 x - 2) (x - 7) (5 x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $\frac{3 x}{(5 x + 2) (x - 7)}$ by $\frac{5 x - 2}{5 x - 2}$ .

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

Step 4.3

Reorder the factors of $(5 x - 2) (x - 7) (5 x + 2)$ .

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

Step 4.4

Reorder the factors of $(5 x + 2) (x - 7) (5 x - 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $2 x (5 x + 2) - 3 x (5 x - 2)$ .

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Step 6.1.1

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

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

Step 6.1.2

Factor $x$ out of $- 3 x (5 x - 2)$ .

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

Step 6.1.3

Factor $x$ out of $x (2 (5 x + 2)) + x (- 3 (5 x - 2))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $5$ by $2$ .

$\frac{x (10 x + 2 \cdot 2 - 3 (5 x - 2))}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.4

Multiply $2$ by $2$ .

$\frac{x (10 x + 4 - 3 (5 x - 2))}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.5

Apply the distributive property.

$\frac{x (10 x + 4 - 3 (5 x) - 3 \cdot - 2)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.6

Multiply $5$ by $- 3$ .

$\frac{x (10 x + 4 - 15 x - 3 \cdot - 2)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.7

Multiply $- 3$ by $- 2$ .

$\frac{x (10 x + 4 - 15 x + 6)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.8

Subtract $15 x$ from $10 x$ .

$\frac{x (- 5 x + 4 + 6)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.9

Add $4$ and $6$ .

$\frac{x (- 5 x + 10)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.10

Factor $5$ out of $- 5 x + 10$ .

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Step 6.10.1

Factor $5$ out of $- 5 x$ .

$\frac{x (5 (- x) + 10)}{(5 x + 2) (5 x - 2) (x - 7)}$

Step 6.10.2

Factor $5$ out of $10$ .

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

Step 6.10.3

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

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

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

Step 7

Simplify with factoring out.

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Step 7.1

Move $5$ to the left of $x$ .

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

Step 7.2

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

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

Step 7.3

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 7.4

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

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

Step 7.5

Simplify the expression.

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Step 7.5.1

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

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

Step 7.5.2

Move the negative in front of the fraction.

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

Step 7.5.3

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

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