Calculus Examples

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

Step 1

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

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

$u = x^{2}$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 8$ to the power of $2$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot - 12}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 12$ .

$u = \frac{8 \pm \sqrt{64 + 48}}{2 \cdot 1}$

Step 4.1.3

Add $64$ and $48$ .

$u = \frac{8 \pm \sqrt{112}}{2 \cdot 1}$

Step 4.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 4.1.4.1

Factor $16$ out of $112$ .

$u = \frac{8 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Step 4.1.4.2

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

$u = \frac{8 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$u = \frac{8 \pm 4 \sqrt{7}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{8 \pm 4 \sqrt{7}}{2}$

Step 4.3

Simplify $\frac{8 \pm 4 \sqrt{7}}{2}$ .

$u = 4 \pm 2 \sqrt{7}$

Step 5

The final answer is the combination of both solutions.

$u = 4 + 2 \sqrt{7}, 4 - 2 \sqrt{7}$

Step 6

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

$x^{2} = 9.29150262$

${(x^{2})}^{1} = - 1.29150262$

Step 7

Solve the first equation for $x$ .

$x^{2} = 9.29150262$

Step 8

Solve the equation for $x$ .

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Step 8.1

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

$x = \pm \sqrt{9.29150262}$

Step 8.2

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

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Step 8.2.1

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

$x = \sqrt{9.29150262}$

Step 8.2.2

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

$x = - \sqrt{9.29150262}$

Step 8.2.3

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

$x = \sqrt{9.29150262}, - \sqrt{9.29150262}$

Step 9

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 1.29150262$

Step 10

Solve the equation for $x$ .

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Step 10.1

Remove parentheses.

$x^{2} = - 1.29150262$

Step 10.2

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

$x = \pm \sqrt{- 1.29150262}$

Step 10.3

Simplify $\pm \sqrt{- 1.29150262}$ .

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Step 10.3.1

Rewrite $- 1.29150262$ as $- 1 (1.29150262)$ .

$x = \pm \sqrt{- 1 (1.29150262)}$

Step 10.3.2

Rewrite $\sqrt{- 1 (1.29150262)}$ as $\sqrt{- 1} \cdot \sqrt{1.29150262}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{1.29150262}$

Step 10.3.3

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

$x = \pm i \sqrt{1.29150262}$

Step 10.4

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

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Step 10.4.1

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

$x = i \sqrt{1.29150262}$

Step 10.4.2

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

$x = - i \sqrt{1.29150262}$

Step 10.4.3

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

$x = i \sqrt{1.29150262}, - i \sqrt{1.29150262}$

Step 11

The solution to $x^{4} - 8 x^{2} - 12 = 0$ is $x = \sqrt{9.29150262}, - \sqrt{9.29150262}, i \sqrt{1.29150262}, - i \sqrt{1.29150262}$ .

$x = \sqrt{9.29150262}, - \sqrt{9.29150262}, i \sqrt{1.29150262}, - i \sqrt{1.29150262}$