Solve by Factoring x = square root of x

$x = \sqrt{x}$

Step 1

Subtract $\sqrt{x}$ from both sides of the equation.

$x - \sqrt{x} = 0$

Step 2

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

$x - x^{\frac{1}{2}} = 0$

Step 3

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

Factor $x^{\frac{1}{2}}$ out of $x^{1}$ .

$x^{\frac{1}{2}} x^{\frac{1}{2}} - x^{\frac{1}{2}} = 0$

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

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

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

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

Step 4

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

$x^{\frac{1}{2}} = 0$

$x^{\frac{1}{2}} - 1 = 0$

Step 5

Set $x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}}$ equal to $0$ .

$x^{\frac{1}{2}} = 0$

Solve $x^{\frac{1}{2}} = 0$ for $x$ .

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the exponent.

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

$x^{1} = 0^{2}$

Simplify.

$x = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 6

Set $x^{\frac{1}{2}} - 1$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 1$ equal to $0$ .

$x^{\frac{1}{2}} - 1 = 0$

Solve $x^{\frac{1}{2}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

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

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x = 1^{2}$

Simplify the right side.

One to any power is one.

$x = 1$

Step 7

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

$x = 0, 1$

Solve by Factoring x = square root of x

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