Find the Domain (x+5)/(x^2+x+12)

$\frac{x + 5}{x^{2} + x + 12}$

Step 1

Set the denominator in $\frac{x + 5}{x^{2} + x + 12}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x + 12 = 0$

Step 2

Solve for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 1$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 12)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12}}{2 \cdot 1}$

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 1}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 1}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12}}{2 \cdot 1}$

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 1}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 1}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{47}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + i \sqrt{47}}{2}$

Factor $- 1$ out of $i \sqrt{47}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{47})}{2}$

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{47})$ .

$x = \frac{- 1 (1 - i \sqrt{47})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{47}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 12}}{2 \cdot 1}$

Multiply $- 4$ by $12$ .

$x = \frac{- 1 \pm \sqrt{1 - 48}}{2 \cdot 1}$

Subtract $48$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 47}}{2 \cdot 1}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{47}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{47}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - i \sqrt{47}}{2}$

Factor $- 1$ out of $- i \sqrt{47}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{47})}{2}$

Factor $- 1$ out of $- 1 (1) - (i \sqrt{47})$ .

$x = \frac{- 1 (1 + i \sqrt{47})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{47}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{47}}{2}, - \frac{1 + i \sqrt{47}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Find the Domain (x+5)/(x^2+x+12)

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