Solve for x x = square root of x

$x = \sqrt{x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{x} = x$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

$x^{1} = x^{2}$

Simplify.

$x = x^{2}$

Step 4

Solve for $x$ .

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.

$x = 0, 1$

Solve for x x = square root of x

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