Calculus Examples

$f (x) = \frac{- 2 x + 8 x^{3}}{2 - x^{2} + 2 x^{4}}$

Step 1

Set the denominator in $\frac{- 2 x + 8 x^{3}}{2 - x^{2} + 2 x^{4}}$ equal to $0$ to find where the expression is undefined.

$2 - x^{2} + 2 x^{4} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$2 u^{2} - u + 2 = 0$

$u = x^{2}$

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 $a = 2$ , $b = - 1$ , and $c = 2$ into the quadratic formula and solve for $u$ .

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

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 $- 1$ to the power of $2$ .

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

Step 2.4.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $2$ .

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

Step 2.4.1.2.2

Multiply $- 8$ by $2$ .

$u = \frac{1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

Step 2.4.1.3

Subtract $16$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 2.4.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 2.4.1.6

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

$u = \frac{1 \pm i \sqrt{15}}{2 \cdot 2}$

Step 2.4.2

Multiply $2$ by $2$ .

$u = \frac{1 \pm i \sqrt{15}}{4}$

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 $- 1$ to the power of $2$ .

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

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

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

Step 2.5.1.2.2

Multiply $- 8$ by $2$ .

$u = \frac{1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

Step 2.5.1.3

Subtract $16$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 2.5.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 2.5.1.6

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

$u = \frac{1 \pm i \sqrt{15}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$u = \frac{1 \pm i \sqrt{15}}{4}$

Step 2.5.3

Change the $\pm$ to $+$ .

$u = \frac{1 + i \sqrt{15}}{4}$

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 $- 1$ to the power of $2$ .

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

Step 2.6.1.2

Multiply $- 4 \cdot 2 \cdot 2$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $2$ .

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

Step 2.6.1.2.2

Multiply $- 8$ by $2$ .

$u = \frac{1 \pm \sqrt{1 - 16}}{2 \cdot 2}$

Step 2.6.1.3

Subtract $16$ from $1$ .

$u = \frac{1 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 2.6.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 2.6.1.6

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

$u = \frac{1 \pm i \sqrt{15}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$u = \frac{1 \pm i \sqrt{15}}{4}$

Step 2.6.3

Change the $\pm$ to $-$ .

$u = \frac{1 - i \sqrt{15}}{4}$

Step 2.7

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{15}}{4}, \frac{1 - i \sqrt{15}}{4}$

Step 2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1 + i \sqrt{15}}{4}$

${(x^{2})}^{1} = \frac{1 - i \sqrt{15}}{4}$

Step 2.9

Solve the first equation for $x$ .

$x^{2} = \frac{1 + i \sqrt{15}}{4}$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1 + i \sqrt{15}}{4}}$

Step 2.10.2

Simplify $\pm \sqrt{\frac{1 + i \sqrt{15}}{4}}$ .

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Step 2.10.2.1

Rewrite $\sqrt{\frac{1 + i \sqrt{15}}{4}}$ as $\frac{\sqrt{1 + i \sqrt{15}}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{15}}}{\sqrt{4}}$

Step 2.10.2.2

Simplify the denominator.

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Step 2.10.2.2.1

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

$x = \pm \frac{\sqrt{1 + i \sqrt{15}}}{\sqrt{2^{2}}}$

Step 2.10.2.2.2

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

$x = \pm \frac{\sqrt{1 + i \sqrt{15}}}{2}$

Step 2.10.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.10.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{1 + i \sqrt{15}}}{2}$

Step 2.10.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{1 + i \sqrt{15}}}{2}$

Step 2.10.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{1 + i \sqrt{15}}}{2}, - \frac{\sqrt{1 + i \sqrt{15}}}{2}$

Step 2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{1 - i \sqrt{15}}{4}$

Step 2.12

Solve the equation for $x$ .

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Step 2.12.1

Remove parentheses.

$x^{2} = \frac{1 - i \sqrt{15}}{4}$

Step 2.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1 - i \sqrt{15}}{4}}$

Step 2.12.3

Simplify $\pm \sqrt{\frac{1 - i \sqrt{15}}{4}}$ .

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Step 2.12.3.1

Rewrite $\sqrt{\frac{1 - i \sqrt{15}}{4}}$ as $\frac{\sqrt{1 - i \sqrt{15}}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{15}}}{\sqrt{4}}$

Step 2.12.3.2

Simplify the denominator.

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Step 2.12.3.2.1

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

$x = \pm \frac{\sqrt{1 - i \sqrt{15}}}{\sqrt{2^{2}}}$

Step 2.12.3.2.2

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

$x = \pm \frac{\sqrt{1 - i \sqrt{15}}}{2}$

Step 2.12.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.12.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{1 - i \sqrt{15}}}{2}$

Step 2.12.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{1 - i \sqrt{15}}}{2}$

Step 2.12.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{1 - i \sqrt{15}}}{2}, - \frac{\sqrt{1 - i \sqrt{15}}}{2}$

Step 2.13

The solution to $2 x^{4} - x^{2} + 2 = 0$ is $x = \frac{\sqrt{1 + i \sqrt{15}}}{2}, - \frac{\sqrt{1 + i \sqrt{15}}}{2}, \frac{\sqrt{1 - i \sqrt{15}}}{2}, - \frac{\sqrt{1 - i \sqrt{15}}}{2}$ .

$x = \frac{\sqrt{1 + i \sqrt{15}}}{2}, - \frac{\sqrt{1 + i \sqrt{15}}}{2}, \frac{\sqrt{1 - i \sqrt{15}}}{2}, - \frac{\sqrt{1 - i \sqrt{15}}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$[- 2.49057894, 2.49057894]$

Set-Builder Notation:

${y | - 2.49057894 \leq y \leq 2.49057894}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[- 2.49057894, 2.49057894], {y | - 2.49057894 \leq y \leq 2.49057894}$

Step 6