Solve 2xfrac{sqrt{2m}}{2-sqrt{2}m}=sqrt{4+2m^2/(sqrt{2m-2)^2}}

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

Solve for x (complex solution)

$$x=-\frac{m^{-\frac{1}{2}}\sqrt{\frac{2\left(m^{2}+4m-4\right)}{m-1}}\left(\sqrt{2}m-2\right)}{4}$$

$$m\neq 0\text{ and }m\neq 1\text{ and }m\neq \sqrt{2}$$

Steps for Solving Linear Equation

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

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{\left(\sqrt{2m-2}\right)^{2}}}$$

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

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{2m-2}}$$

Factor the expressions that are not already factored in \frac{2m^{2}}{2m-2}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{2\left(m-1\right)}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{m^{2}}{m-1}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{m-1}{m-1}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4\left(m-1\right)}{m-1}+\frac{m^{2}}{m-1}}$$

Since \frac{4\left(m-1\right)}{m-1} and \frac{m^{2}}{m-1} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4\left(m-1\right)+m^{2}}{m-1}}$$

Do the multiplications in 4\left(m-1\right)+m^{2}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Factor the expressions that are not already factored in \frac{2\sqrt{2m}}{2-\sqrt{2}m}.

$$\frac{2\sqrt{2m}}{\sqrt{2}\left(-m+\sqrt{2}\right)}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Cancel out \sqrt{2} in both numerator and denominator.

$$\frac{\sqrt{2}\sqrt{2m}}{-m+\sqrt{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Rationalize the denominator of \frac{\sqrt{2}\sqrt{2m}}{-m+\sqrt{2}} by multiplying numerator and denominator by -m-\sqrt{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-m+\sqrt{2}\right)\left(-m-\sqrt{2}\right)}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Consider \left(-m+\sqrt{2}\right)\left(-m-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-m\right)^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Expand \left(-m\right)^{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-1\right)^{2}m^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-2}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Express \frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-2}x as a single fraction.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)x}{1m^{2}-2}=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Multiply both sides of the equation by m^{2}-2.

$$\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Use the distributive property to multiply \sqrt{2}\sqrt{2m} by -m-\sqrt{2}.

$$\left(-\sqrt{2}\sqrt{2m}m-\sqrt{2m}\left(\sqrt{2}\right)^{2}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The square of \sqrt{2} is 2.

$$\left(-\sqrt{2}\sqrt{2m}m-\sqrt{2m}\times 2\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Multiply -1 and 2 to get -2.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

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

$$-\sqrt{2}\sqrt{2m}mx-2\sqrt{2m}x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Use the distributive property to multiply m^{2}-2 by \sqrt{\frac{4m-4+m^{2}}{m-1}}.

$$-\sqrt{2}\sqrt{2m}mx-2\sqrt{2m}x=m^{2}\sqrt{\frac{4m-4+m^{2}}{m-1}}-2\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Combine all terms containing x.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=m^{2}\sqrt{\frac{4m-4+m^{2}}{m-1}}-2\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The equation is in standard form.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=\sqrt{\frac{m^{2}+4m-4}{m-1}}m^{2}-2\sqrt{\frac{m^{2}+4m-4}{m-1}}$$

Divide both sides by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$\frac{\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}=\frac{\sqrt{\frac{\left(m+2\right)^{2}-8}{m-1}}\left(m^{2}-2\right)}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}$$

Dividing by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m} undoes the multiplication by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$x=\frac{\sqrt{\frac{\left(m+2\right)^{2}-8}{m-1}}\left(m^{2}-2\right)}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}$$

Divide \left(m^{2}-2\right)\sqrt{\frac{\left(2+m\right)^{2}-8}{m-1}} by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$x=-\frac{m^{-\frac{1}{2}}\sqrt{\frac{m^{2}+4m-4}{m-1}}\left(m^{2}-2\right)}{2\left(m+\sqrt{2}\right)}$$

Solve for x

$$x=-\frac{\sqrt{\frac{2\left(m^{2}+4m-4\right)}{m\left(m-1\right)}}\left(\sqrt{2}m-2\right)}{4}$$

$$m>1\text{ and }m\neq \sqrt{2}$$

Steps for Solving Linear Equation

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

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{\left(\sqrt{2m-2}\right)^{2}}}$$

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

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{2m-2}}$$

Factor the expressions that are not already factored in \frac{2m^{2}}{2m-2}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{2m^{2}}{2\left(m-1\right)}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{4+\frac{m^{2}}{m-1}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{m-1}{m-1}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4\left(m-1\right)}{m-1}+\frac{m^{2}}{m-1}}$$

Since \frac{4\left(m-1\right)}{m-1} and \frac{m^{2}}{m-1} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4\left(m-1\right)+m^{2}}{m-1}}$$

Do the multiplications in 4\left(m-1\right)+m^{2}.

$$\frac{2\sqrt{2m}}{2-\sqrt{2}m}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Factor the expressions that are not already factored in \frac{2\sqrt{2m}}{2-\sqrt{2}m}.

$$\frac{2\sqrt{2m}}{\sqrt{2}\left(-m+\sqrt{2}\right)}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Cancel out \sqrt{2} in both numerator and denominator.

$$\frac{\sqrt{2}\sqrt{2m}}{-m+\sqrt{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Rationalize the denominator of \frac{\sqrt{2}\sqrt{2m}}{-m+\sqrt{2}} by multiplying numerator and denominator by -m-\sqrt{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-m+\sqrt{2}\right)\left(-m-\sqrt{2}\right)}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Consider \left(-m+\sqrt{2}\right)\left(-m-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-m\right)^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Expand \left(-m\right)^{2}.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{\left(-1\right)^{2}m^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-\left(\sqrt{2}\right)^{2}}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-2}x=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Express \frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)}{1m^{2}-2}x as a single fraction.

$$\frac{\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)x}{1m^{2}-2}=\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Multiply both sides of the equation by m^{2}-2.

$$\sqrt{2}\sqrt{2m}\left(-m-\sqrt{2}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Use the distributive property to multiply \sqrt{2}\sqrt{2m} by -m-\sqrt{2}.

$$\left(-\sqrt{2}\sqrt{2m}m-\sqrt{2m}\left(\sqrt{2}\right)^{2}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The square of \sqrt{2} is 2.

$$\left(-\sqrt{2}\sqrt{2m}m-\sqrt{2m}\times 2\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Multiply -1 and 2 to get -2.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

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

$$-\sqrt{2}\sqrt{2m}mx-2\sqrt{2m}x=\left(m^{2}-2\right)\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Use the distributive property to multiply m^{2}-2 by \sqrt{\frac{4m-4+m^{2}}{m-1}}.

$$-\sqrt{2}\sqrt{2m}mx-2\sqrt{2m}x=m^{2}\sqrt{\frac{4m-4+m^{2}}{m-1}}-2\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

Combine all terms containing x.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=m^{2}\sqrt{\frac{4m-4+m^{2}}{m-1}}-2\sqrt{\frac{4m-4+m^{2}}{m-1}}$$

The equation is in standard form.

$$\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x=\sqrt{\frac{m^{2}+4m-4}{m-1}}m^{2}-2\sqrt{\frac{m^{2}+4m-4}{m-1}}$$

Divide both sides by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$\frac{\left(-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}\right)x}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}=\frac{\sqrt{\frac{\left(m+2-2\sqrt{2}\right)\left(m+2\sqrt{2}+2\right)}{m-1}}\left(m^{2}-2\right)}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}$$

Dividing by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m} undoes the multiplication by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$x=\frac{\sqrt{\frac{\left(m+2-2\sqrt{2}\right)\left(m+2\sqrt{2}+2\right)}{m-1}}\left(m^{2}-2\right)}{-\sqrt{2}\sqrt{2m}m-2\sqrt{2m}}$$

Divide \left(m^{2}-2\right)\sqrt{\frac{\left(2+2\sqrt{2}+m\right)\left(2-2\sqrt{2}+m\right)}{m-1}} by -\sqrt{2}\sqrt{2m}m-2\sqrt{2m}.

$$x=-\frac{\sqrt{\frac{2\left(m+2-2\sqrt{2}\right)\left(m+2\sqrt{2}+2\right)}{m-1}}\left(m^{2}-2\right)}{2\sqrt{m}\left(\sqrt{2}m+2\right)}$$