Pre-Algebra Examples

$(\sqrt{x + 4} + x^{2}) (12)$

Step 1

Reorder $12 \sqrt{x + 4}$ and $12 x^{2}$ .

$y = 12 x^{2} + 12 \sqrt{x + 4}$

Step 2

Find the domain for $y = (\sqrt{x + 4} + x^{2}) (12)$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 2.1

Set the radicand in $\sqrt{x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$x + 4 \geq 0$

Step 2.2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 2.3

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

Interval Notation:

$[- 4, \infty)$

Set-Builder Notation:

${x | x \geq - 4}$

Interval Notation:

$[- 4, \infty)$

Set-Builder Notation:

${x | x \geq - 4}$

Step 3

To find the radical expression end point, substitute the $x$ value $- 4$ , which is the least value in the domain, into $f (x) = 12 x^{2} + 12 \sqrt{x + 4}$ .

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Step 3.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = 12 {(- 4)}^{2} + 12 \sqrt{(- 4) + 4}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = 12 \cdot 16 + 12 \sqrt{(- 4) + 4}$

Step 3.2.1.2

Multiply $12$ by $16$ .

$f (- 4) = 192 + 12 \sqrt{(- 4) + 4}$

Step 3.2.1.3

Add $- 4$ and $4$ .

$f (- 4) = 192 + 12 \sqrt{0}$

Step 3.2.1.4

Rewrite $0$ as $0^{2}$ .

$f (- 4) = 192 + 12 \sqrt{0^{2}}$

Step 3.2.1.5

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

$f (- 4) = 192 + 12 \cdot 0$

Step 3.2.1.6

Multiply $12$ by $0$ .

$f (- 4) = 192 + 0$

Step 3.2.2

Add $192$ and $0$ .

$f (- 4) = 192$

Step 3.2.3

The final answer is $192$ .

$192$

Step 4

The radical expression end point is $(- 4, 192)$ .

$(- 4, 192)$

Step 5

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 5.1

Substitute the $x$ value $- 3$ into $f (x) = 12 x^{2} + 12 \sqrt{x + 4}$ . In this case, the point is $(- 3, 120)$ .

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Step 5.1.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 12 {(- 3)}^{2} + 12 \sqrt{(- 3) + 4}$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = 12 \cdot 9 + 12 \sqrt{(- 3) + 4}$

Step 5.1.2.1.2

Multiply $12$ by $9$ .

$f (- 3) = 108 + 12 \sqrt{(- 3) + 4}$

Step 5.1.2.1.3

Add $- 3$ and $4$ .

$f (- 3) = 108 + 12 \sqrt{1}$

Step 5.1.2.1.4

Any root of $1$ is $1$ .

$f (- 3) = 108 + 12 \cdot 1$

Step 5.1.2.1.5

Multiply $12$ by $1$ .

$f (- 3) = 108 + 12$

Step 5.1.2.2

Add $108$ and $12$ .

$f (- 3) = 120$

Step 5.1.2.3

The final answer is $120$ .

$y = 120$

Step 5.2

Substitute the $x$ value $- 2$ into $f (x) = 12 x^{2} + 12 \sqrt{x + 4}$ . In this case, the point is $(- 2, 48 + 12 \sqrt{2})$ .

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Step 5.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 12 {(- 2)}^{2} + 12 \sqrt{(- 2) + 4}$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

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

$f (- 2) = 12 \cdot 4 + 12 \sqrt{(- 2) + 4}$

Step 5.2.2.1.2

Multiply $12$ by $4$ .

$f (- 2) = 48 + 12 \sqrt{(- 2) + 4}$

Step 5.2.2.1.3

Add $- 2$ and $4$ .

$f (- 2) = 48 + 12 \sqrt{2}$

Step 5.2.2.2

The final answer is $48 + 12 \sqrt{2}$ .

$y = 48 + 12 \sqrt{2}$

Step 5.3

The square root can be graphed using the points around the vertex $(- 4, 192), (- 3, 120), (- 2, 64.97)$

$\begin{array}{c} x & y \\ - 4 & 192 \\ - 3 & 120 \\ - 2 & 64.97 \end{array}$

Step 6