Algebra Examples

$\frac{3}{4 x^{2}} - \frac{x}{2 x^{2} + 10 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $2 x$ out of $2 x^{2} + 10 x$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $2 x$ out of $10 x$ .

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

Step 1.1.3

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

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

$\frac{3}{4 x^{2}} - \frac{1}{2 (x + 5)}$

Step 2

To write $\frac{3}{4 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3}{4 x^{2}} \cdot \frac{x + 5}{x + 5} - \frac{1}{2 (x + 5)}$

Step 3

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

$\frac{3}{4 x^{2}} \cdot \frac{x + 5}{x + 5} - \frac{1}{2 (x + 5)} \cdot \frac{2 x^{2}}{2 x^{2}}$

Step 4

Write each expression with a common denominator of $4 (x + 5) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3}{4 x^{2}}$ by $\frac{x + 5}{x + 5}$ .

$\frac{3 (x + 5)}{4 x^{2} (x + 5)} - \frac{1}{2 (x + 5)} \cdot \frac{2 x^{2}}{2 x^{2}}$

Step 4.2

Multiply $\frac{1}{2 (x + 5)}$ by $\frac{2 x^{2}}{2 x^{2}}$ .

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

Step 4.3

Multiply $2$ by $2$ .

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

Step 4.4

Reorder the factors of $4 (x + 5) x^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $5$ .

$\frac{3 x + 15 - 2 x^{2}}{4 x^{2} (x + 5)}$

Step 6.3

Reorder terms.

$\frac{- 2 x^{2} + 3 x + 15}{4 x^{2} (x + 5)}$

Step 7

Simplify with factoring out.

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Step 7.1

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

$\frac{- (2 x^{2}) + 3 x + 15}{4 x^{2} (x + 5)}$

Step 7.2

Factor $- 1$ out of $3 x$ .

$\frac{- (2 x^{2}) - (- 3 x) + 15}{4 x^{2} (x + 5)}$

Step 7.3

Factor $- 1$ out of $- (2 x^{2}) - (- 3 x)$ .

$\frac{- (2 x^{2} - 3 x) + 15}{4 x^{2} (x + 5)}$

Step 7.4

Rewrite $15$ as $- 1 (- 15)$ .

$\frac{- (2 x^{2} - 3 x) - 1 (- 15)}{4 x^{2} (x + 5)}$

Step 7.5

Factor $- 1$ out of $- (2 x^{2} - 3 x) - 1 (- 15)$ .

$\frac{- (2 x^{2} - 3 x - 15)}{4 x^{2} (x + 5)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (2 x^{2} - 3 x - 15)$ as $- 1 (2 x^{2} - 3 x - 15)$ .

$\frac{- 1 (2 x^{2} - 3 x - 15)}{4 x^{2} (x + 5)}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{2 x^{2} - 3 x - 15}{4 x^{2} (x + 5)}$