Calculus Examples

$n (x) = \frac{x - 2}{x^{2} - 2 x - 3}$

Step 1

Divide each term in $n x = \frac{x - 2}{x^{2} - 2 x - 3}$ by $x$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $n x = \frac{x - 2}{x^{2} - 2 x - 3}$ by $x$ .

$\frac{n x}{x} = \frac{\frac{x - 2}{x^{2} - 2 x - 3}}{x}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{n x}{x} = \frac{\frac{x - 2}{x^{2} - 2 x - 3}}{x}$

Step 1.2.1.2

Divide $n$ by $1$ .

$n = \frac{\frac{x - 2}{x^{2} - 2 x - 3}}{x}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3.2

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

Tap for more steps...

Step 1.3.2.1

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 1.3.2.2

Write the factored form using these integers.

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

Step 1.3.3

Combine fractions.

Tap for more steps...

Step 1.3.3.1

Multiply $\frac{x - 2}{(x - 3) (x + 1)}$ by $\frac{1}{x}$ .

$n = \frac{x - 2}{(x - 3) (x + 1) x}$

Step 1.3.3.2

Reorder factors in $\frac{x - 2}{(x - 3) (x + 1) x}$ .

$n = \frac{x - 2}{x (x - 3) (x + 1)}$

Step 2

To rewrite as a function of $x$ , write the equation so that $n$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{x - 2}{x (x - 3) (x + 1)}$