Algebra Examples

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

Step 1

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

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

$u = x^{2}$

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 $a = 1$ , $b = - 4$ , and $c = 2$ into the quadratic formula and solve for $u$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 4$ to the power of $2$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 2}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $2$ .

$u = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Step 4.1.3

Subtract $8$ from $16$ .

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

Step 4.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.4.1

Factor $4$ out of $8$ .

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

Step 4.1.4.2

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

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

Step 4.1.5

Pull terms out from under the radical.

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

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

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

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

Step 5

The final answer is the combination of both solutions.

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

Step 6

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

$x^{2} = 3.41421356$

${(x^{2})}^{1} = 0.58578643$

Step 7

Solve the first equation for $x$ .

$x^{2} = 3.41421356$

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.

$x = \pm \sqrt{3.41421356}$

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.

$x = \sqrt{3.41421356}$

Step 8.2.2

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

$x = - \sqrt{3.41421356}$

Step 8.2.3

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

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}$

Step 9

Solve the second equation for $x$ .

${(x^{2})}^{1} = 0.58578643$

Step 10

Solve the equation for $x$ .

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Step 10.1

Remove parentheses.

$x^{2} = 0.58578643$

Step 10.2

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

$x = \pm \sqrt{0.58578643}$

Step 10.3

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

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Step 10.3.1

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

$x = \sqrt{0.58578643}$

Step 10.3.2

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

$x = - \sqrt{0.58578643}$

Step 10.3.3

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

$x = \sqrt{0.58578643}, - \sqrt{0.58578643}$

Step 11

The solution to $x^{4} - 4 x^{2} + 2 = 0$ is $x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$ .

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3.41421356}, - \sqrt{3.41421356}, \sqrt{0.58578643}, - \sqrt{0.58578643}$

Decimal Form:

$x = 1.84775906 \dots, - 1.84775906 \dots, 0.76536686 \dots, - 0.76536686 \dots$