Precalculus Examples

$r (t) = \frac{t^{2} - 1}{t^{3} + 6 t^{2} + t}$

Step 1

Set the denominator in $\frac{t^{2} - 1}{t^{3} + 6 t^{2} + t}$ equal to $0$ to find where the expression is undefined.

$t^{3} + 6 t^{2} + t = 0$

Step 2

Solve for $t$ .

Tap for more steps...

Step 2.1

Factor $t$ out of $t^{3} + 6 t^{2} + t$ .

Tap for more steps...

Step 2.1.1

Factor $t$ out of $t^{3}$ .

$t \cdot t^{2} + 6 t^{2} + t = 0$

Step 2.1.2

Factor $t$ out of $6 t^{2}$ .

$t \cdot t^{2} + t (6 t) + t = 0$

Step 2.1.3

Raise $t$ to the power of $1$ .

$t \cdot t^{2} + t (6 t) + t = 0$

Step 2.1.4

Factor $t$ out of $t^{1}$ .

$t \cdot t^{2} + t (6 t) + t \cdot 1 = 0$

Step 2.1.5

Factor $t$ out of $t \cdot t^{2} + t (6 t)$ .

$t (t^{2} + 6 t) + t \cdot 1 = 0$

Step 2.1.6

Factor $t$ out of $t (t^{2} + 6 t) + t \cdot 1$ .

$t (t^{2} + 6 t + 1) = 0$

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 $0$ .

$t = 0$

$t^{2} + 6 t + 1 = 0$

Step 2.3

Set $t$ equal to $0$ .

$t = 0$

Step 2.4

Set $t^{2} + 6 t + 1$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 2.4.1

Set $t^{2} + 6 t + 1$ equal to $0$ .

$t^{2} + 6 t + 1 = 0$

Step 2.4.2

Solve $t^{2} + 6 t + 1 = 0$ for $t$ .

Tap for more steps...

Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 1$ into the quadratic formula and solve for $t$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

Tap for more steps...

Step 2.4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.3.1.1

Raise $6$ to the power of $2$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.3.1.2

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

Tap for more steps...

Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $4$ from $36$ .

$t = \frac{- 6 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.3.1.4.1

Factor $16$ out of $32$ .

$t = \frac{- 6 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.4.2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 6 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Pull terms out from under the radical.

$t = \frac{- 6 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$t = \frac{- 6 \pm 4 \sqrt{2}}{2}$

Step 2.4.2.3.3

Simplify $\frac{- 6 \pm 4 \sqrt{2}}{2}$ .

$t = - 3 \pm 2 \sqrt{2}$

Step 2.4.2.4

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

Tap for more steps...

Step 2.4.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.4.1.1

Raise $6$ to the power of $2$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.4.1.2

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

Tap for more steps...

Step 2.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.4.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4}}{2 \cdot 1}$

Step 2.4.2.4.1.3

Subtract $4$ from $36$ .

$t = \frac{- 6 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.4.2.4.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.4.1.4.1

Factor $16$ out of $32$ .

$t = \frac{- 6 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.4.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 6 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.4.2.4.1.5

Pull terms out from under the radical.

$t = \frac{- 6 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.4.2.4.2

Multiply $2$ by $1$ .

$t = \frac{- 6 \pm 4 \sqrt{2}}{2}$

Step 2.4.2.4.3

Simplify $\frac{- 6 \pm 4 \sqrt{2}}{2}$ .

$t = - 3 \pm 2 \sqrt{2}$

Step 2.4.2.4.4

Change the $\pm$ to $+$ .

$t = - 3 + 2 \sqrt{2}$

Step 2.4.2.5

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

Tap for more steps...

Step 2.4.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.5.1.1

Raise $6$ to the power of $2$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.5.1.2

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

Tap for more steps...

Step 2.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.4.2.5.1.2.2

Multiply $- 4$ by $1$ .

$t = \frac{- 6 \pm \sqrt{36 - 4}}{2 \cdot 1}$

Step 2.4.2.5.1.3

Subtract $4$ from $36$ .

$t = \frac{- 6 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.4.2.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.5.1.4.1

Factor $16$ out of $32$ .

$t = \frac{- 6 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.4.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 6 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.4.2.5.1.5

Pull terms out from under the radical.

$t = \frac{- 6 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.4.2.5.2

Multiply $2$ by $1$ .

$t = \frac{- 6 \pm 4 \sqrt{2}}{2}$

Step 2.4.2.5.3

Simplify $\frac{- 6 \pm 4 \sqrt{2}}{2}$ .

$t = - 3 \pm 2 \sqrt{2}$

Step 2.4.2.5.4

Change the $\pm$ to $-$ .

$t = - 3 - 2 \sqrt{2}$

Step 2.4.2.6

The final answer is the combination of both solutions.

$t = - 3 + 2 \sqrt{2}, - 3 - 2 \sqrt{2}$

Step 2.5

The final solution is all the values that make $t (t^{2} + 6 t + 1) = 0$ true.

$t = 0, - 3 + 2 \sqrt{2}, - 3 - 2 \sqrt{2}$

Step 3

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

Interval Notation:

$(- \infty, - 3 - 2 \sqrt{2}) \cup (- 3 - 2 \sqrt{2}, - 3 + 2 \sqrt{2}) \cup (- 3 + 2 \sqrt{2}, 0) \cup (0, \infty)$

Set-Builder Notation:

${t | t \neq 0, - 3 + 2 \sqrt{2}, - 3 - 2 \sqrt{2}}$

Step 4