Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $15 x^{2}$ .

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{3}{x^{2}} - \frac{1}{3}}{\frac{x + 2}{15} - \frac{1}{x}}$ by $\frac{15 x^{2}}{15 x^{2}}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $15$ by $3$ .

$\frac{45 + 15 x^{2} (- \frac{1}{3})}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{45 + 15 x^{2} \frac{- 1}{3}}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.3.2

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

$\frac{45 + 3 (5 x^{2}) \frac{- 1}{3}}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.3.3

Cancel the common factor.

$\frac{45 + 3 (5 x^{2}) \frac{- 1}{3}}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.3.4

Rewrite the expression.

$\frac{45 + 5 x^{2} \cdot - 1}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.4

Multiply $- 1$ by $5$ .

$\frac{45 - 5 x^{2}}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.5

Cancel the common factor of $15$ .

Tap for more steps...

Step 3.5.1

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

$\frac{45 - 5 x^{2}}{15 (x^{2}) \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.5.2

Cancel the common factor.

$\frac{45 - 5 x^{2}}{15 x^{2} \frac{x + 2}{15} + 15 x^{2} (- \frac{1}{x})}$

Step 3.5.3

Rewrite the expression.

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) + 15 x^{2} (- \frac{1}{x})}$

Step 3.6

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) + 15 x^{2} \frac{- 1}{x}}$

Step 3.6.2

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

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) + x (15 x) \frac{- 1}{x}}$

Step 3.6.3

Cancel the common factor.

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) + x (15 x) \frac{- 1}{x}}$

Step 3.6.4

Rewrite the expression.

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) + 15 x \cdot - 1}$

Step 3.7

Multiply $- 1$ by $15$ .

$\frac{45 - 5 x^{2}}{x^{2} (x + 2) - 15 x}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Factor $5$ out of $45 - 5 x^{2}$ .

Tap for more steps...

Step 4.1.1

Factor $5$ out of $45$ .

$\frac{5 (9) - 5 x^{2}}{x^{2} (x + 2) - 15 x}$

Step 4.1.2

Factor $5$ out of $- 5 x^{2}$ .

$\frac{5 (9) + 5 (- x^{2})}{x^{2} (x + 2) - 15 x}$

Step 4.1.3

Factor $5$ out of $5 (9) + 5 (- x^{2})$ .

$\frac{5 (9 - x^{2})}{x^{2} (x + 2) - 15 x}$

Step 4.2

Rewrite $9$ as $3^{2}$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Factor $x$ out of $x^{2} (x + 2)$ .

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

Step 5.1.2

Factor $x$ out of $- 15 x$ .

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

Step 5.1.3

Factor $x$ out of $x (x (x + 2)) + x \cdot - 15$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x$ .

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

Step 5.4

Move $2$ to the left of $x$ .

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

Step 5.5

Factor $x^{2} + 2 x - 15$ using the AC method.

Tap for more steps...

Step 5.5.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 5.5.2

Write the factored form using these integers.

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

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

Step 6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.1

Cancel the common factor of $3 - x$ and $x - 3$ .

Tap for more steps...

Step 6.1.1

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

Tap for more steps...

Step 6.2.1

Multiply $- 1$ by $5$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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