Find the Roots (Zeros) f(x)=x^4-2x^2

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

Step 1

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

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

Step 2

Solve for $x$ .

Factor the left side of the equation.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 2 x^{2} = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

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

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

$u \cdot u - 2 u = 0$

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

$u \cdot u + u \cdot - 2 = 0$

Factor $u$ out of $u \cdot u + u \cdot - 2$ .

$u (u - 2) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 2 = 0$

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Solve $x^{2} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{2} = 2$

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

$x = \pm \sqrt{2}$

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

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

$x = \sqrt{2}$

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

$x = - \sqrt{2}$

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

$x = \sqrt{2}, - \sqrt{2}$

The final solution is all the values that make $x^{2} (x^{2} - 2) = 0$ true.

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 0, 1.41421356 \dots, - 1.41421356 \dots$

Step 4

Find the Roots (Zeros) f(x)=x^4-2x^2

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