Calculus Examples

$f (x) = x^{4} - 8 x^{2} + 5$

Step 1

Set $x^{4} - 8 x^{2} + 5$ equal to $0$ .

$x^{4} - 8 x^{2} + 5 = 0$

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.

$u^{2} - 8 u + 5 = 0$

$u = x^{2}$

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 $a = 1$ , $b = - 8$ , and $c = 5$ into the quadratic formula and solve for $u$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

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

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 5}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $5$ .

$u = \frac{8 \pm \sqrt{64 - 20}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $20$ from $64$ .

$u = \frac{8 \pm \sqrt{44}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 2.4.1.4.1

Factor $4$ out of $44$ .

$u = \frac{8 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 2.4.1.4.2

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

$u = \frac{8 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 2.4.1.5

Pull terms out from under the radical.

$u = \frac{8 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$u = \frac{8 \pm 2 \sqrt{11}}{2}$

Step 2.4.3

Simplify $\frac{8 \pm 2 \sqrt{11}}{2}$ .

$u = 4 \pm \sqrt{11}$

Step 2.5

The final answer is the combination of both solutions.

$u = 4 + \sqrt{11}, 4 - \sqrt{11}$

Step 2.6

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

$x^{2} = 7.31662479$

${(x^{2})}^{1} = 0.6833752$

Step 2.7

Solve the first equation for $x$ .

$x^{2} = 7.31662479$

Step 2.8

Solve the equation for $x$ .

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Step 2.8.1

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

$x = \pm \sqrt{7.31662479}$

Step 2.8.2

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

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Step 2.8.2.1

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

$x = \sqrt{7.31662479}$

Step 2.8.2.2

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

$x = - \sqrt{7.31662479}$

Step 2.8.2.3

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

$x = \sqrt{7.31662479}, - \sqrt{7.31662479}$

Step 2.9

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.6833752$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Remove parentheses.

$x^{2} = 0.6833752$

Step 2.10.2

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

$x = \pm \sqrt{0.6833752}$

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.

$x = \sqrt{0.6833752}$

Step 2.10.3.2

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

$x = - \sqrt{0.6833752}$

Step 2.10.3.3

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

$x = \sqrt{0.6833752}, - \sqrt{0.6833752}$

Step 2.11

The solution to $x^{4} - 8 x^{2} + 5 = 0$ is $x = \sqrt{7.31662479}, - \sqrt{7.31662479}, \sqrt{0.6833752}, - \sqrt{0.6833752}$ .

$x = \sqrt{7.31662479}, - \sqrt{7.31662479}, \sqrt{0.6833752}, - \sqrt{0.6833752}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{7.31662479}, - \sqrt{7.31662479}, \sqrt{0.6833752}, - \sqrt{0.6833752}$

Decimal Form:

$x = 2.70492602 \dots, - 2.70492602 \dots, 0.82666511 \dots, - 0.82666511 \dots$

Step 4