Algebra Examples

$x^{4} + 12 x^{2} - 8$

Step 1

Set $x^{4} + 12 x^{2} - 8$ equal to $0$ .

$x^{4} + 12 x^{2} - 8 = 0$

Step 2

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

$u^{2} + 12 u - 8 = 0$

$u = x^{2}$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = 12$ , and $c = - 8$ into the quadratic formula and solve for $u$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $12$ to the power of $2$ .

$u = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 8}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 8$ .

$u = \frac{- 12 \pm \sqrt{144 + 32}}{2 \cdot 1}$

Step 5.1.3

Add $144$ and $32$ .

$u = \frac{- 12 \pm \sqrt{176}}{2 \cdot 1}$

Step 5.1.4

Rewrite $176$ as $4^{2} \cdot 11$ .

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Step 5.1.4.1

Factor $16$ out of $176$ .

$u = \frac{- 12 \pm \sqrt{16 (11)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $16$ as $4^{2}$ .

$u = \frac{- 12 \pm \sqrt{4^{2} \cdot 11}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$u = \frac{- 12 \pm 4 \sqrt{11}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$u = \frac{- 12 \pm 4 \sqrt{11}}{2}$

Step 5.3

Simplify $\frac{- 12 \pm 4 \sqrt{11}}{2}$ .

$u = - 6 \pm 2 \sqrt{11}$

Step 6

The final answer is the combination of both solutions.

$u = - 6 + 2 \sqrt{11}, - 6 - 2 \sqrt{11}$

Step 7

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

$x^{2} = 0.63324958$

${(x^{2})}^{1} = - 12.63324958$

Step 8

Solve the first equation for $x$ .

$x^{2} = 0.63324958$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x = \pm \sqrt{0.63324958}$

Step 9.2

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

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Step 9.2.1

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

$x = \sqrt{0.63324958}$

Step 9.2.2

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

$x = - \sqrt{0.63324958}$

Step 9.2.3

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

$x = \sqrt{0.63324958}, - \sqrt{0.63324958}$

Step 10

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 12.63324958$

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = - 12.63324958$

Step 11.2

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

$x = \pm \sqrt{- 12.63324958}$

Step 11.3

Simplify $\pm \sqrt{- 12.63324958}$ .

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Step 11.3.1

Rewrite $- 12.63324958$ as $- 1 (12.63324958)$ .

$x = \pm \sqrt{- 1 (12.63324958)}$

Step 11.3.2

Rewrite $\sqrt{- 1 (12.63324958)}$ as $\sqrt{- 1} \cdot \sqrt{12.63324958}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{12.63324958}$

Step 11.3.3

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

$x = \pm i \sqrt{12.63324958}$

Step 11.4

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

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Step 11.4.1

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

$x = i \sqrt{12.63324958}$

Step 11.4.2

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

$x = - i \sqrt{12.63324958}$

Step 11.4.3

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

$x = i \sqrt{12.63324958}, - i \sqrt{12.63324958}$

Step 12

The solution to $x^{4} + 12 x^{2} - 8 = 0$ is $x = \sqrt{0.63324958}, - \sqrt{0.63324958}, i \sqrt{12.63324958}, - i \sqrt{12.63324958}$ .

$x = \sqrt{0.63324958}, - \sqrt{0.63324958}, i \sqrt{12.63324958}, - i \sqrt{12.63324958}$