Algebra Examples

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

Step 1

Factor $x^{2}$ out of $x^{4} - x^{2}$ .

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Step 1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.2

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

$x^{2} x^{2} + x^{2} \cdot - 1 = 0$

Step 1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 1$ .

$x^{2} (x^{2} - 1) = 0$

Step 2

Rewrite $1$ as $1^{2}$ .

$x^{2} (x^{2} - 1^{2}) = 0$

Step 3

Factor.

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Step 3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$x^{2} ((x + 1) (x - 1)) = 0$

Step 3.2

Remove unnecessary parentheses.

$x^{2} (x + 1) (x - 1) = 0$

Step 4

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

$x^{2} = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 5

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

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Step 5.1

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

$x^{2} = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.1

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

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

Step 5.2.2.2

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

$x = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 8

The final solution is all the values that make $x^{2} (x + 1) (x - 1) = 0$ true.

$x = 0, - 1, 1$