Trigonometry Examples

$\frac{\frac{3 x^{2} + 12 x - 15}{x^{2} + 2 x - 15}}{\frac{3 x^{2} - 3 x}{3 x - x^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x^{2} + 12 x - 15}{x^{2} + 2 x - 15} \cdot \frac{3 x - x^{2}}{3 x^{2} - 3 x}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 x^{2} + 12 x - 15$ .

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Step 2.1.1

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

$\frac{3 (x^{2}) + 12 x - 15}{x^{2} + 2 x - 15} \cdot \frac{3 x - x^{2}}{3 x^{2} - 3 x}$

Step 2.1.2

Factor $3$ out of $12 x$ .

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

Step 2.1.3

Factor $3$ out of $- 15$ .

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

Step 2.1.4

Factor $3$ out of $3 x^{2} + 3 (4 x)$ .

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

Step 2.1.5

Factor $3$ out of $3 (x^{2} + 4 x) + 3 \cdot - 5$ .

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

Step 2.2

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 2.2.1

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 $4$ .

$- 1, 5$

Step 2.2.2

Write the factored form using these integers.

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

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

Step 3

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

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Step 3.1

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 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 5$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

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

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Step 4.2.1

Factor $x$ out of $3 x$ .

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

Step 4.2.2

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

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

Step 4.2.3

Factor $x$ out of $x \cdot 3 + x (- x)$ .

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Factor $3 x$ out of $- 3 x$ .

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

Step 4.3.3

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

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

Step 4.4

Cancel the common factor of $3 (x - 1)$ .

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

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

Step 4.5

Multiply $\frac{1}{x - 3}$ by $\frac{x (3 - x)}{x}$ .

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

Step 4.6

Cancel the common factor of $x$ .

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Step 4.6.1

Cancel the common factor.

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

Step 4.6.2

Rewrite the expression.

$\frac{3 - x}{x - 3}$

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Reorder terms.

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

Step 4.7.5

Cancel the common factor.

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

Step 4.7.6

Divide $- 1$ by $1$ .

$- 1$