Algebra Examples

$\frac{\frac{1}{x^{2}} - 1}{\frac{x - 4}{5} - \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $5 x^{2}$ .

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{1}{x^{2}} - 1}{\frac{x - 4}{5} - \frac{1}{x}}$ by $\frac{5 x^{2}}{5 x^{2}}$ .

$\frac{5 x^{2}}{5 x^{2}} \cdot \frac{\frac{1}{x^{2}} - 1}{\frac{x - 4}{5} - \frac{1}{x}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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 $5 x^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Multiply $- 1$ by $5$ .

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Multiply $- 1$ by $5$ .

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

Step 4.2

Rewrite $5 - 5 x^{2}$ in a factored form.

Tap for more steps...

Step 4.2.1

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

Tap for more steps...

Step 4.2.1.1

Factor $5$ out of $5$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Factor $5$ out of $5 (1) + 5 (- x^{2})$ .

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

Step 4.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 4.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Factor $x$ out of $x^{2} (x - 4) - 5 x$ .

Tap for more steps...

Step 5.1.1

Factor $x$ out of $x^{2} (x - 4)$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x$ out of $x (x (x - 4)) + x \cdot - 5$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x$ .

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

Step 5.4

Move $- 4$ to the left of $x$ .

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

Step 5.5

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 5.5.2

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $1 + x$ and $x + 1$ .

Tap for more steps...

Step 6.1

Reorder terms.

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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