Solve sqrt{2}*(x/sqrt{2}-sqrt{3})=x-sqrt{6}

$$\sqrt { 2 } \times ( \frac { x } { \sqrt { 2 } } - \sqrt { 3 } ) = x - \sqrt { 6 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply \sqrt{2} by \frac{x\sqrt{2}}{2}-\sqrt{3}.

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

Express \sqrt{2}\times \frac{x\sqrt{2}}{2} as a single fraction.

$$\frac{\sqrt{2}x\sqrt{2}}{2}-\sqrt{2}\sqrt{3}=x-\sqrt{6}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

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

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

Cancel out 2 and 2.

$$x-\sqrt{6}=x-\sqrt{6}$$

Subtract x from both sides.

$$x-\sqrt{6}-x=-\sqrt{6}$$

Combine x and -x to get 0.

$$-\sqrt{6}=-\sqrt{6}$$

Cancel out -1 on both sides.

$$\sqrt{6}=\sqrt{6}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply \sqrt{2} by \frac{x\sqrt{2}}{2}-\sqrt{3}.

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

Express \sqrt{2}\times \frac{x\sqrt{2}}{2} as a single fraction.

$$\frac{\sqrt{2}x\sqrt{2}}{2}-\sqrt{2}\sqrt{3}=x-\sqrt{6}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

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

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

Cancel out 2 and 2.

$$x-\sqrt{6}=x-\sqrt{6}$$

Subtract x from both sides.

$$x-\sqrt{6}-x=-\sqrt{6}$$

Combine x and -x to get 0.

$$-\sqrt{6}=-\sqrt{6}$$

Cancel out -1 on both sides.

$$\sqrt{6}=\sqrt{6}$$

This is true for any x.

$$x\in \mathrm{R}$$