Trigonometry Examples

$\frac{7 x - x^{2}}{x^{2} - 49}$

Step 1

Factor $x$ out of $7 x - x^{2}$ .

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Step 1.1

Factor $x$ out of $7 x$ .

$\frac{x \cdot 7 - x^{2}}{x^{2} - 49}$

Step 1.2

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

$\frac{x \cdot 7 + x (- x)}{x^{2} - 49}$

Step 1.3

Factor $x$ out of $x \cdot 7 + x (- x)$ .

$\frac{x (7 - x)}{x^{2} - 49}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{x (7 - x)}{x^{2} - 7^{2}}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{x (7 - x)}{(x + 7) (x - 7)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $7 - x$ and $x - 7$ .

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Step 3.1.1

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{x (- 1 (- 7) - x)}{(x + 7) (x - 7)}$

Step 3.1.2

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

$\frac{x (- 1 (- 7) - (x))}{(x + 7) (x - 7)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- 7) - (x)$ .

$\frac{x (- 1 (- 7 + x))}{(x + 7) (x - 7)}$

Step 3.1.4

Reorder terms.

$\frac{x (- 1 (x - 7))}{(x + 7) (x - 7)}$

Step 3.1.5

Cancel the common factor.

$\frac{x (- 1 (x - 7))}{(x + 7) (x - 7)}$

Step 3.1.6

Rewrite the expression.

$\frac{x \cdot (- 1)}{x + 7}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{x + 7}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{x}{x + 7}$