Solve frac{sqrt{2}x}{sqrt{6}}=sqrt{x}

$$\frac { \sqrt { 2 } x } { \sqrt { 6 } } = \sqrt { x }$$

Solve for x

$$x=0$$

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\frac{\sqrt{2}x}{\sqrt{6}}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Rationalize the denominator of \frac{\sqrt{2}x}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\left(\frac{\sqrt{2}x\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

The square of \sqrt{6} is 6.

$$\left(\frac{\sqrt{2}x\sqrt{6}}{6}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\left(\frac{\sqrt{2}x\sqrt{2}\sqrt{3}}{6}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\left(\frac{2x\sqrt{3}}{6}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Divide 2x\sqrt{3} by 6 to get \frac{1}{3}x\sqrt{3}.

$$\left(\frac{1}{3}x\sqrt{3}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Expand \left(\frac{1}{3}x\sqrt{3}\right)^{2}.

$$\left(\frac{1}{3}\right)^{2}x^{2}\left(\sqrt{3}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{1}{9}x^{2}\left(\sqrt{3}\right)^{2}=\left(\sqrt{x}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{1}{9}x^{2}\times 3=\left(\sqrt{x}\right)^{2}$$

Multiply \frac{1}{9} and 3 to get \frac{3}{9}.

$$\frac{3}{9}x^{2}=\left(\sqrt{x}\right)^{2}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$\frac{1}{3}x^{2}=\left(\sqrt{x}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{1}{3}x^{2}=x$$

Subtract x from both sides.

$$\frac{1}{3}x^{2}-x=0$$

Factor out x.

$$x\left(\frac{1}{3}x-1\right)=0$$

To find equation solutions, solve x=0 and \frac{x}{3}-1=0.

$$x=0 x=3$$

Substitute 0 for x in the equation \frac{\sqrt{2}x}{\sqrt{6}}=\sqrt{x}.

$$\frac{\sqrt{2}\times 0}{\sqrt{6}}=\sqrt{0}$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Substitute 3 for x in the equation \frac{\sqrt{2}x}{\sqrt{6}}=\sqrt{x}.

$$\frac{\sqrt{2}\times 3}{\sqrt{6}}=\sqrt{3}$$

Simplify. The value x=3 satisfies the equation.

$$3^{\frac{1}{2}}=3^{\frac{1}{2}}$$

List all solutions of \frac{\sqrt{2}x}{\sqrt{6}}=\sqrt{x}.

$$x=0 x=3$$