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

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

Solve for x

$$x = \frac{18 \sqrt{2} + 6}{17} \approx 1.850343772$$

Steps for Solving Linear Equation

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

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

Consider \left(6-\sqrt{2}\right)\left(6+\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}.

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

Square 6. Square \sqrt{2}.

$$x=\frac{6\sqrt{2}\left(6+\sqrt{2}\right)}{36-2}$$

Subtract 2 from 36 to get 34.

$$x=\frac{6\sqrt{2}\left(6+\sqrt{2}\right)}{34}$$

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

$$x=\frac{36\sqrt{2}+6\left(\sqrt{2}\right)^{2}}{34}$$

The square of \sqrt{2} is 2.

$$x=\frac{36\sqrt{2}+6\times 2}{34}$$

Multiply 6 and 2 to get 12.

$$x=\frac{36\sqrt{2}+12}{34}$$

Divide each term of 36\sqrt{2}+12 by 34 to get \frac{18}{17}\sqrt{2}+\frac{6}{17}.

$$x=\frac{18}{17}\sqrt{2}+\frac{6}{17}$$

Assign x

$$x≔\frac{18\sqrt{2}+6}{17}$$