Finite Math Examples

$- \sqrt{x^{2} + 8 x + 32}$

Step 1

Set the radicand in $\sqrt{x^{2} + 8 x + 32}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 8 x + 32 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$x^{2} + 8 x + 32 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 1$ , $b = 8$ , and $c = 32$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot 32)}}{2 \cdot 1}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 32}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 32$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 32}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $32$ .

$x = \frac{- 8 \pm \sqrt{64 - 128}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $128$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 64}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.4.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.4.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 2.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 8 \pm i \cdot 8}{2 \cdot 1}$

Step 2.4.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i}{2}$

Step 2.4.3

Simplify $\frac{- 8 \pm 8 i}{2}$ .

$x = - 4 \pm 4 i$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 32}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 32$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 32}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $32$ .

$x = \frac{- 8 \pm \sqrt{64 - 128}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $128$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 64}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.5.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.5.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 2.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 8 \pm i \cdot 8}{2 \cdot 1}$

Step 2.5.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i}{2}$

Step 2.5.3

Simplify $\frac{- 8 \pm 8 i}{2}$ .

$x = - 4 \pm 4 i$

Step 2.5.4

Change the $\pm$ to $+$ .

$x = - 4 + 4 i$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 32}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 32$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 32}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $32$ .

$x = \frac{- 8 \pm \sqrt{64 - 128}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $128$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 64}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.6.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 2.6.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 2.6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 8 \pm i \cdot 8}{2 \cdot 1}$

Step 2.6.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i}{2}$

Step 2.6.3

Simplify $\frac{- 8 \pm 8 i}{2}$ .

$x = - 4 \pm 4 i$

Step 2.6.4

Change the $\pm$ to $-$ .

$x = - 4 - 4 i$

Step 2.7

Identify the leading coefficient.

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Step 2.7.1

The leading term in a polynomial is the term with the highest degree.

$x^{2}$

Step 2.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 2.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} + 8 x + 32$ is always greater than $0$ .

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4