Find the Domain (x^2+5x+6)/(x^2-3x-10)

$\frac{x^{2} + 5 x + 6}{x^{2} - 3 x - 10}$

Step 1

Set the denominator in $\frac{x^{2} + 5 x + 6}{x^{2} - 3 x - 10}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 3 x - 10 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(x - 5) (x + 2) = 0$

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

$x - 5 = 0$

$x + 2 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

Step 3

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, 5) \cup (5, \infty)$

Set-Builder Notation:

${x | x \neq - 2, 5}$

Step 4

Find the Domain (x^2+5x+6)/(x^2-3x-10)

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