Find the Domain y=(x^2-5x-6)/(x^2-3x-18)

$y = \frac{x^{2} - 5 x - 6}{x^{2} - 3 x - 18}$

Step 1

Set the denominator in $\frac{x^{2} - 5 x - 6}{x^{2} - 3 x - 18}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 3 x - 18 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} - 3 x - 18$ using the AC method.

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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Write the factored form using these integers.

$(x - 6) (x + 3) = 0$

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

$x - 6 = 0$

$x + 3 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 6) (x + 3) = 0$ true.

$x = 6, - 3$

Step 3

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq - 3, 6}$

Step 4

Find the Domain y=(x^2-5x-6)/(x^2-3x-18)

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