Solve the Rational Equation for x square root of 2x-4 = square root of x

$\sqrt{2 x - 4} = \sqrt{x}$

Step 1

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

${\sqrt{2 x - 4}}^{2} = {\sqrt{x}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 4}$ as ${(2 x - 4)}^{\frac{1}{2}}$ .

${({(2 x - 4)}^{\frac{1}{2}})}^{2} = {\sqrt{x}}^{2}$

Simplify the left side.

Simplify ${({(2 x - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x - 4)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x - 4)}^{\frac{1}{2} \cdot 2} = {\sqrt{x}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x - 4)}^{\frac{1}{2} \cdot 2} = {\sqrt{x}}^{2}$

Rewrite the expression.

${(2 x - 4)}^{1} = {\sqrt{x}}^{2}$

Simplify.

$2 x - 4 = {\sqrt{x}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

$2 x - 4 = {(x^{\frac{1}{2}})}^{2}$

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

$2 x - 4 = x^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$2 x - 4 = x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x - 4 = x^{\frac{2}{2}}$

Rewrite the expression.

$2 x - 4 = x^{1}$

Simplify.

$2 x - 4 = x$

Step 3

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$2 x - 4 - x = 0$

Subtract $x$ from $2 x$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$