Solve 4x/sqrt{2+sqrt{6}}=4/sqrt{2}

$$\frac { 4 x } { \sqrt { 2 } + \sqrt { 6 } } = \frac { 4 } { \sqrt { 2 } }$$

Solve for x

$$x=\sqrt{3}+1\approx 2.732050808$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{4x}{\sqrt{2}+\sqrt{6}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{6}.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right)}=\frac{4}{\sqrt{2}}$$

Consider \left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}=\frac{4}{\sqrt{2}}$$

Square \sqrt{2}. Square \sqrt{6}.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{2-6}=\frac{4}{\sqrt{2}}$$

Subtract 6 from 2 to get -4.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{-4}=\frac{4}{\sqrt{2}}$$

Rationalize the denominator of \frac{4}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{-4}=\frac{4\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{-4}=\frac{4\sqrt{2}}{2}$$

Divide 4\sqrt{2} by 2 to get 2\sqrt{2}.

$$\frac{4x\left(\sqrt{2}-\sqrt{6}\right)}{-4}=2\sqrt{2}$$

Cancel out -4 and -4.

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

Use the distributive property to multiply -x by \sqrt{2}-\sqrt{6}.

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

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by -\sqrt{2}+\sqrt{6}.

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

Dividing by -\sqrt{2}+\sqrt{6} undoes the multiplication by -\sqrt{2}+\sqrt{6}.

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

Divide 2\sqrt{2} by -\sqrt{2}+\sqrt{6}.

$$x=\sqrt{3}+1$$