Calculus Examples

$f (x) = \frac{2 x^{2} - 4 x + 2}{3 x^{2} - 3 x - 6}$

Step 1

Set the denominator in $\frac{2 x^{2} - 4 x + 2}{3 x^{2} - 3 x - 6}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} - 3 x - 6 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $3$ out of $3 x^{2} - 3 x - 6$ .

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Step 2.1.1.1

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

$3 (x^{2}) - 3 x - 6 = 0$

Step 2.1.1.2

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

$3 (x^{2}) + 3 (- x) - 6 = 0$

Step 2.1.1.3

Factor $3$ out of $- 6$ .

$3 (x^{2}) + 3 (- x) + 3 (- 2) = 0$

Step 2.1.1.4

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

$3 (x^{2} - x) + 3 (- 2) = 0$

Step 2.1.1.5

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

$3 (x^{2} - x - 2) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 2.1.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 1)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 1) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 1 = 0$

Step 2.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

The final solution is all the values that make $3 (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 2}$

Step 4