Precalculus Examples

$\frac{3 x^{2} + 1}{x^{2} + x + 9}$

Step 1

Set the denominator in $\frac{3 x^{2} + 1}{x^{2} + x + 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x + 9 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 1 \pm \sqrt{1 - 36}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $36$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 35}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $- 35$ as $- 1 (35)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 35}}{2 \cdot 1}$

Step 2.3.1.5

Rewrite $\sqrt{- 1 (35)}$ as $\sqrt{- 1} \cdot \sqrt{35}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{35}}{2 \cdot 1}$

Step 2.3.1.6

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

$x = \frac{- 1 \pm i \sqrt{35}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{35}}{2}$

Step 2.4

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

Tap for more steps...

Step 2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 1 \pm \sqrt{1 - 36}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $36$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 35}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $- 35$ as $- 1 (35)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 35}}{2 \cdot 1}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (35)}$ as $\sqrt{- 1} \cdot \sqrt{35}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{35}}{2 \cdot 1}$

Step 2.4.1.6

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

$x = \frac{- 1 \pm i \sqrt{35}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{35}}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{35}}{2}$

Step 2.4.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{35}}{2}$

Step 2.4.5

Factor $- 1$ out of $i \sqrt{35}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{35})}{2}$

Step 2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{35})$ .

$x = \frac{- 1 (1 - i \sqrt{35})}{2}$

Step 2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{35}}{2}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

Tap for more steps...

Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 1 \pm \sqrt{1 - 36}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $36$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 35}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $- 35$ as $- 1 (35)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 35}}{2 \cdot 1}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (35)}$ as $\sqrt{- 1} \cdot \sqrt{35}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{35}}{2 \cdot 1}$

Step 2.5.1.6

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

$x = \frac{- 1 \pm i \sqrt{35}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{35}}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{35}}{2}$

Step 2.5.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{35}}{2}$

Step 2.5.5

Factor $- 1$ out of $- i \sqrt{35}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{35})}{2}$

Step 2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{35})$ .

$x = \frac{- 1 (1 + i \sqrt{35})}{2}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{35}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{35}}{2}, - \frac{1 + i \sqrt{35}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4