Trigonometry Examples

$\frac{\frac{x}{10} - \frac{1}{10} - \frac{2}{x}}{\frac{1}{2} - \frac{x}{10}}$

Step 1

Multiply the numerator and denominator of the fraction by $10 x$ .

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{x}{10} - \frac{1}{10} - \frac{2}{x}}{\frac{1}{2} - \frac{x}{10}}$ by $\frac{10 x}{10 x}$ .

$\frac{10 x}{10 x} \cdot \frac{\frac{x}{10} - \frac{1}{10} - \frac{2}{x}}{\frac{1}{2} - \frac{x}{10}}$

Step 1.2

Combine.

$\frac{10 x (\frac{x}{10} - \frac{1}{10} - \frac{2}{x})}{10 x (\frac{1}{2} - \frac{x}{10})}$

Step 2

Apply the distributive property.

$\frac{10 x \frac{x}{10} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.1.1

Factor $10$ out of $10 x$ .

$\frac{10 (x) \frac{x}{10} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.1.2

Cancel the common factor.

$\frac{10 x \frac{x}{10} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.1.3

Rewrite the expression.

$\frac{x \cdot x + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.2

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

$\frac{x^{1} x + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.3

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

$\frac{x^{1} x^{1} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.4

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

$\frac{x^{1 + 1} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.5

Add $1$ and $1$ .

$\frac{x^{2} + 10 x (- \frac{1}{10}) + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.6

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.6.1

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$\frac{x^{2} + 10 x \frac{- 1}{10} + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.6.2

Factor $10$ out of $10 x$ .

$\frac{x^{2} + 10 (x) \frac{- 1}{10} + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.6.3

Cancel the common factor.

$\frac{x^{2} + 10 x \frac{- 1}{10} + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.6.4

Rewrite the expression.

$\frac{x^{2} + x \cdot - 1 + 10 x (- \frac{2}{x})}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.7

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.7.1

Move the leading negative in $- \frac{2}{x}$ into the numerator.

$\frac{x^{2} + x \cdot - 1 + 10 x \frac{- 2}{x}}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.7.2

Factor $x$ out of $10 x$ .

$\frac{x^{2} + x \cdot - 1 + x \cdot 10 \frac{- 2}{x}}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.7.3

Cancel the common factor.

$\frac{x^{2} + x \cdot - 1 + x \cdot 10 \frac{- 2}{x}}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.7.4

Rewrite the expression.

$\frac{x^{2} + x \cdot - 1 + 10 \cdot - 2}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.8

Multiply $10$ by $- 2$ .

$\frac{x^{2} + x \cdot - 1 - 20}{10 x \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.9.1

Factor $2$ out of $10 x$ .

$\frac{x^{2} + x \cdot - 1 - 20}{2 (5 x) \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.9.2

Cancel the common factor.

$\frac{x^{2} + x \cdot - 1 - 20}{2 (5 x) \frac{1}{2} + 10 x (- \frac{x}{10})}$

Step 3.9.3

Rewrite the expression.

$\frac{x^{2} + x \cdot - 1 - 20}{5 x + 10 x (- \frac{x}{10})}$

Step 3.10

Cancel the common factor of $10$ .

Tap for more steps...

Step 3.10.1

Move the leading negative in $- \frac{x}{10}$ into the numerator.

$\frac{x^{2} + x \cdot - 1 - 20}{5 x + 10 x \frac{- x}{10}}$

Step 3.10.2

Factor $10$ out of $10 x$ .

$\frac{x^{2} + x \cdot - 1 - 20}{5 x + 10 (x) \frac{- x}{10}}$

Step 3.10.3

Cancel the common factor.

$\frac{x^{2} + x \cdot - 1 - 20}{5 x + 10 x \frac{- x}{10}}$

Step 3.10.4

Rewrite the expression.

$\frac{x^{2} + x \cdot - 1 - 20}{5 x + x (- x)}$

Step 3.11

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

$\frac{x^{2} + x \cdot - 1 - 20}{5 x - (x^{1} x)}$

Step 3.12

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

$\frac{x^{2} + x \cdot - 1 - 20}{5 x - (x^{1} x^{1})}$

Step 3.13

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

$\frac{x^{2} + x \cdot - 1 - 20}{5 x - x^{1 + 1}}$

Step 3.14

Add $1$ and $1$ .

$\frac{x^{2} + x \cdot - 1 - 20}{5 x - x^{2}}$

Step 4

Factor $x^{2} + x \cdot - 1 - 20$ using the AC method.

Tap for more steps...

Step 4.1

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Factor $x$ out of $5 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Cancel the common factor of $x - 5$ and $5 - x$ .

Tap for more steps...

Step 5.2.1

Factor $- 1$ out of $x$ .

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

Step 5.2.2

Rewrite $- 5$ as $- 1 (5)$ .

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

Step 5.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

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

Step 5.2.4

Reorder terms.

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

Step 5.2.5

Cancel the common factor.

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

Step 5.2.6

Rewrite the expression.

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

Step 5.3

Move the negative in front of the fraction.

$- \frac{x + 4}{x}$