Finite Math Examples

$y = \sqrt{x^{3} + 3 x^{2} + 3 x} - \sqrt{x^{3} + 3 x^{2} + 3 x}$

Step 1

Set the radicand in $\sqrt{x^{3} + 3 x^{2} + 3 x}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} + 3 x^{2} + 3 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$x^{3} + 3 x^{2} + 3 x = 0$

Step 2.2

Factor $x$ out of $x^{3} + 3 x^{2} + 3 x$ .

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Step 2.2.1

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

$x \cdot x^{2} + 3 x^{2} + 3 x = 0$

Step 2.2.2

Factor $x$ out of $3 x^{2}$ .

$x \cdot x^{2} + x (3 x) + 3 x = 0$

Step 2.2.3

Factor $x$ out of $3 x$ .

$x \cdot x^{2} + x (3 x) + x \cdot 3 = 0$

Step 2.2.4

Factor $x$ out of $x \cdot x^{2} + x (3 x)$ .

$x (x^{2} + 3 x) + x \cdot 3 = 0$

Step 2.2.5

Factor $x$ out of $x (x^{2} + 3 x) + x \cdot 3$ .

$x (x^{2} + 3 x + 3) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} + 3 x + 3 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} + 3 x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + 3 x + 3$ equal to $0$ .

$x^{2} + 3 x + 3 = 0$

Step 2.5.2

Solve $x^{2} + 3 x + 3 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Subtract $12$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.3.1.6

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

$x = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{3}}{2}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Step 2.5.2.4.1.3

Subtract $12$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.4.1.6

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

$x = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{3}}{2}$

Step 2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{3}}{2}$

Step 2.5.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{3}}{2}$

Step 2.5.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{3})}{2}$

Step 2.5.2.4.6

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{3})$ .

$x = \frac{- 1 (3 - i \sqrt{3})}{2}$

Step 2.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{3}}{2}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Step 2.5.2.5.1.3

Subtract $12$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.5.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.5.1.6

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

$x = \frac{- 3 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{3}}{2}$

Step 2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{3}}{2}$

Step 2.5.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{3}}{2}$

Step 2.5.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{3})}{2}$

Step 2.5.2.5.6

Factor $- 1$ out of $- 1 (3) - (i \sqrt{3})$ .

$x = \frac{- 1 (3 + i \sqrt{3})}{2}$

Step 2.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{3}}{2}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{3}}{2}, - \frac{3 + i \sqrt{3}}{2}$

Step 2.6

The final solution is all the values that make $x (x^{2} + 3 x + 3) = 0$ true.

$x = 0, - \frac{3 - i \sqrt{3}}{2}, - \frac{3 + i \sqrt{3}}{2}$

Step 2.7

The solution consists of all of the true intervals.

$x \geq 0$

Step 3

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, 0]$

Set-Builder Notation:

${y | y = 0}$

Step 5

Determine the domain and range.

Domain: $[0, \infty), {x | x \geq 0}$

Range: $[0, 0], {y | y = 0}$

Step 6