Algebra Examples

$\frac{7}{2 x^{2} + 18 x} - \frac{5 x}{x^{2} + 17 x + 72}$

Step 1

Simplify each term.

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Step 1.1

Factor $2 x$ out of $2 x^{2} + 18 x$ .

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Step 1.1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{7}{2 x (x) + 18 x} - \frac{5 x}{x^{2} + 17 x + 72}$

Step 1.1.2

Factor $2 x$ out of $18 x$ .

$\frac{7}{2 x (x) + 2 x (9)} - \frac{5 x}{x^{2} + 17 x + 72}$

Step 1.1.3

Factor $2 x$ out of $2 x (x) + 2 x (9)$ .

$\frac{7}{2 x (x + 9)} - \frac{5 x}{x^{2} + 17 x + 72}$

Step 1.2

Factor $x^{2} + 17 x + 72$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $72$ and whose sum is $17$ .

$8, 9$

Step 1.2.2

Write the factored form using these integers.

$\frac{7}{2 x (x + 9)} - \frac{5 x}{(x + 8) (x + 9)}$

Step 2

To write $\frac{7}{2 x (x + 9)}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{7}{2 x (x + 9)} \cdot \frac{x + 8}{x + 8} - \frac{5 x}{(x + 8) (x + 9)}$

Step 3

To write $- \frac{5 x}{(x + 8) (x + 9)}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{7}{2 x (x + 9)} \cdot \frac{x + 8}{x + 8} - \frac{5 x}{(x + 8) (x + 9)} \cdot \frac{2 x}{2 x}$

Step 4

Write each expression with a common denominator of $2 x (x + 9) (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{7}{2 x (x + 9)}$ by $\frac{x + 8}{x + 8}$ .

$\frac{7 (x + 8)}{2 x (x + 9) (x + 8)} - \frac{5 x}{(x + 8) (x + 9)} \cdot \frac{2 x}{2 x}$

Step 4.2

Multiply $\frac{5 x}{(x + 8) (x + 9)}$ by $\frac{2 x}{2 x}$ .

$\frac{7 (x + 8)}{2 x (x + 9) (x + 8)} - \frac{5 x (2 x)}{(x + 8) (x + 9) (2 x)}$

Step 4.3

Reorder the factors of $2 x (x + 9) (x + 8)$ .

$\frac{7 (x + 8)}{2 x (x + 8) (x + 9)} - \frac{5 x (2 x)}{(x + 8) (x + 9) (2 x)}$

Step 4.4

Reorder the factors of $(x + 8) (x + 9) (2 x)$ .

$\frac{7 (x + 8)}{2 x (x + 8) (x + 9)} - \frac{5 x (2 x)}{2 x (x + 8) (x + 9)}$

Step 5

Combine the numerators over the common denominator.

$\frac{7 (x + 8) - 5 x (2 x)}{2 x (x + 8) (x + 9)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{7 x + 7 \cdot 8 - 5 x (2 x)}{2 x (x + 8) (x + 9)}$

Step 6.2

Multiply $7$ by $8$ .

$\frac{7 x + 56 - 5 x (2 x)}{2 x (x + 8) (x + 9)}$

Step 6.3

Rewrite using the commutative property of multiplication.

$\frac{7 x + 56 - 5 \cdot 2 x \cdot x}{2 x (x + 8) (x + 9)}$

Step 6.4

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.1

Move $x$ .

$\frac{7 x + 56 - 5 \cdot 2 (x \cdot x)}{2 x (x + 8) (x + 9)}$

Step 6.4.2

Multiply $x$ by $x$ .

$\frac{7 x + 56 - 5 \cdot 2 x^{2}}{2 x (x + 8) (x + 9)}$

Step 6.5

Multiply $- 5$ by $2$ .

$\frac{7 x + 56 - 10 x^{2}}{2 x (x + 8) (x + 9)}$

Step 6.6

Reorder terms.

$\frac{- 10 x^{2} + 7 x + 56}{2 x (x + 8) (x + 9)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 10 x^{2}$ .

$\frac{- (10 x^{2}) + 7 x + 56}{2 x (x + 8) (x + 9)}$

Step 7.2

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

$\frac{- (10 x^{2}) - (- 7 x) + 56}{2 x (x + 8) (x + 9)}$

Step 7.3

Factor $- 1$ out of $- (10 x^{2}) - (- 7 x)$ .

$\frac{- (10 x^{2} - 7 x) + 56}{2 x (x + 8) (x + 9)}$

Step 7.4

Rewrite $56$ as $- 1 (- 56)$ .

$\frac{- (10 x^{2} - 7 x) - 1 (- 56)}{2 x (x + 8) (x + 9)}$

Step 7.5

Factor $- 1$ out of $- (10 x^{2} - 7 x) - 1 (- 56)$ .

$\frac{- (10 x^{2} - 7 x - 56)}{2 x (x + 8) (x + 9)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (10 x^{2} - 7 x - 56)$ as $- 1 (10 x^{2} - 7 x - 56)$ .

$\frac{- 1 (10 x^{2} - 7 x - 56)}{2 x (x + 8) (x + 9)}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{10 x^{2} - 7 x - 56}{2 x (x + 8) (x + 9)}$