Trigonometry Examples

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

Step 1

Simplify $\frac{a}{3 x} \cdot \frac{3 x + 3}{x^{2} - 2 x - 3}$ .

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Step 1.1

Factor $3$ out of $3 x + 3$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

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

Step 1.1.2

Factor $3$ out of $3$ .

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

Step 1.1.3

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

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

Step 1.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 1.2.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Cancel the common factor of $3$ .

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Step 1.3.1.1

Factor $3$ out of $3 x$ .

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

Step 1.3.1.2

Cancel the common factor.

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

Step 1.3.1.3

Rewrite the expression.

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

Step 1.3.2

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

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

Step 1.3.3

Cancel the common factor of $x + 1$ .

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Step 1.3.3.1

Cancel the common factor.

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

Step 1.3.3.2

Rewrite the expression.

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

Step 2

Multiply both sides by $x (x - 3)$ .

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

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

Cancel the common factor.

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

Step 3.1.1.2

Rewrite the expression.

$a = 1 (x (x - 3))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $1 (x (x - 3))$ .

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Step 3.2.1.1

Multiply $x (x - 3)$ by $1$ .

$a = x (x - 3)$

Step 3.2.1.2

Apply the distributive property.

$a = x \cdot x + x \cdot - 3$

Step 3.2.1.3

Simplify the expression.

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Step 3.2.1.3.1

Multiply $x$ by $x$ .

$a = x^{2} + x \cdot - 3$

Step 3.2.1.3.2

Move $- 3$ to the left of $x$ .

$a = x^{2} - 3 x$