Find All Complex Solutions x^2=|x|

$x^{2} = | x |$

Step 1

Rewrite the equation as $| x | = x^{2}$ .

$| x | = x^{2}$

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm x^{2}$

Step 3

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.

$x = x^{2}$

Subtract $x^{2}$ from both sides of the equation.

$x - x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - u^{2} = 0$

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

Raise $u$ to the power of $1$ .

$u - u^{2} = 0$

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - u^{2} = 0$

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

$u \cdot 1 + u (- u) = 0$

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$u (1 - u) = 0$

Replace all occurrences of $u$ with $x$ .

$x (1 - x) = 0$

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

$x = 0$

$1 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$1 - x = 0$

Solve $1 - x = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

The final solution is all the values that make $x (1 - x) = 0$ true.

$x = 0, 1$

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

$x = - x^{2}$

Add $x^{2}$ to both sides of the equation.

$x + x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u + u^{2} = 0$

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

Raise $u$ to the power of $1$ .

$u + u^{2} = 0$

Factor $u$ out of $u^{1}$ .

$u \cdot 1 + u^{2} = 0$

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

$u \cdot 1 + u \cdot u = 0$

Factor $u$ out of $u \cdot 1 + u \cdot u$ .

$u (1 + u) = 0$

Replace all occurrences of $u$ with $x$ .

$x (1 + x) = 0$

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

$x = 0$

$1 + x = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$1 + x = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $x (1 + x) = 0$ true.

$x = 0, - 1$

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

$x = 0, 1, - 1$

Find All Complex Solutions x^2=|x|

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