Algebra Examples

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

Step 1

Factor $3 x$ out of $3 x^{2} + 15 x$ .

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Step 1.1

Factor $3 x$ out of $3 x^{2}$ .

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

Step 1.2

Factor $3 x$ out of $15 x$ .

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

Step 1.3

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

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

Step 2

To write $\frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{3 (x + 5)}{3 (x + 5)}$ .

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $3 (x + 5) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{x^{2}}$ by $\frac{3 (x + 5)}{3 (x + 5)}$ .

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

Step 4.2

Multiply $\frac{4}{3 x (x + 5)}$ by $\frac{x}{x}$ .

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

Step 4.3

Raise $x$ to the power of $1$ .

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

Step 4.4

Raise $x$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Reorder the factors of $x^{2} (3 (x + 5))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $5$ .

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

Step 6.3

Subtract $4 x$ from $3 x$ .

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

Step 7

Simplify with factoring out.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Factor $- 1$ out of $- (x) - 1 (- 15)$ .

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

Step 7.4

Simplify the expression.

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Step 7.4.1

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

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

Step 7.4.2

Move the negative in front of the fraction.

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