Find All Complex Solutions square root of x^2=x

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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

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

Divide $2$ by $2$ .

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

Simplify the left side.

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

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

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

Multiply $2$ by $1$ .

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

Step 3

Solve for $x$ .

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | x |$

Solve for $x$ .

Rewrite the absolute value equation as four equations without absolute value bars.

$x = x$

$x = - x$

$- x = x$

$- x = - x$

After simplifying, there are only two unique equations to be solved.

$x = x$

$x = - x$

Solve $x = x$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

Solve $x = - x$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$x + x = 0$

Add $x$ and $x$ .

$2 x = 0$

Divide each term in $2 x = 0$ by $2$ and simplify.

Divide each term in $2 x = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $0$ by $2$ .

$x = 0$

List all of the solutions.

$x = 0$

Step 4

Verify each of the solutions by substituting them into $\sqrt{x^{2}} = x$ and solving.

$x = 0$

Find All Complex Solutions square root of x^2=x

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