Calculus Examples

$- 6 x^{4} - 6 x^{2} + 2 = 0$

Step 1

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

$- 6 u^{2} - 6 u + 2 = 0$

$u = x^{2}$

Step 2

Factor $- 2$ out of $- 6 u^{2} - 6 u + 2$ .

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Step 2.1

Factor $- 2$ out of $- 6 u^{2}$ .

$- 2 (3 u^{2}) - 6 u + 2 = 0$

Step 2.2

Factor $- 2$ out of $- 6 u$ .

$- 2 (3 u^{2}) - 2 (3 u) + 2 = 0$

Step 2.3

Factor $- 2$ out of $2$ .

$- 2 (3 u^{2}) - 2 (3 u) - 2 \cdot - 1 = 0$

Step 2.4

Factor $- 2$ out of $- 2 (3 u^{2}) - 2 (3 u)$ .

$- 2 (3 u^{2} + 3 u) - 2 \cdot - 1 = 0$

Step 2.5

Factor $- 2$ out of $- 2 (3 u^{2} + 3 u) - 2 (- 1)$ .

$- 2 (3 u^{2} + 3 u - 1) = 0$

Step 3

Divide each term in $- 2 (3 u^{2} + 3 u - 1) = 0$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 (3 u^{2} + 3 u - 1) = 0$ by $- 2$ .

$\frac{- 2 (3 u^{2} + 3 u - 1)}{- 2} = \frac{0}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 (3 u^{2} + 3 u - 1)}{- 2} = \frac{0}{- 2}$

Step 3.2.1.2

Divide $3 u^{2} + 3 u - 1$ by $1$ .

$3 u^{2} + 3 u - 1 = \frac{0}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 2$ .

$3 u^{2} + 3 u - 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

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

Step 6.1.2.2

Multiply $- 12$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 12}}{2 \cdot 3}$

Step 6.1.3

Add $9$ and $12$ .

$u = \frac{- 3 \pm \sqrt{21}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$u = \frac{- 3 \pm \sqrt{21}}{6}$

Step 7

The final answer is the combination of both solutions.

$u = - \frac{3 - \sqrt{21}}{6}, - \frac{3 + \sqrt{21}}{6}$

Step 8

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

$x^{2} = 0.26376261$

${(x^{2})}^{1} = - 1.26376261$

Step 9

Solve the first equation for $x$ .

$x^{2} = 0.26376261$

Step 10

Solve the equation for $x$ .

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Step 10.1

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

$x = \pm \sqrt{0.26376261}$

Step 10.2

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

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Step 10.2.1

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

$x = \sqrt{0.26376261}$

Step 10.2.2

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

$x = - \sqrt{0.26376261}$

Step 10.2.3

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

$x = \sqrt{0.26376261}, - \sqrt{0.26376261}$

Step 11

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 1.26376261$

Step 12

Solve the equation for $x$ .

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Step 12.1

Remove parentheses.

$x^{2} = - 1.26376261$

Step 12.2

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

$x = \pm \sqrt{- 1.26376261}$

Step 12.3

Simplify $\pm \sqrt{- 1.26376261}$ .

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Step 12.3.1

Rewrite $- 1.26376261$ as $- 1 (1.26376261)$ .

$x = \pm \sqrt{- 1 (1.26376261)}$

Step 12.3.2

Rewrite $\sqrt{- 1 (1.26376261)}$ as $\sqrt{- 1} \cdot \sqrt{1.26376261}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{1.26376261}$

Step 12.3.3

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

$x = \pm i \sqrt{1.26376261}$

Step 12.4

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

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Step 12.4.1

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

$x = i \sqrt{1.26376261}$

Step 12.4.2

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

$x = - i \sqrt{1.26376261}$

Step 12.4.3

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

$x = i \sqrt{1.26376261}, - i \sqrt{1.26376261}$

Step 13

The solution to $- 6 x^{4} - 6 x^{2} + 2 = 0$ is $x = \sqrt{0.26376261}, - \sqrt{0.26376261}, i \sqrt{1.26376261}, - i \sqrt{1.26376261}$ .

$x = \sqrt{0.26376261}, - \sqrt{0.26376261}, i \sqrt{1.26376261}, - i \sqrt{1.26376261}$