Solve 6+x^2left(1+2sqrt{2}right)=sqrt{2}left(2x^2-6xsqrt{3}+3sqrt{2}right)

$$6+ { x }^{ 2 } \left( 1+2 \sqrt{ 2 } \right) = \sqrt{ 2 } \left( 2 { x }^{ 2 } -6x \sqrt{ 3 } +3 \sqrt{ 2 } \right)$$

Solve for x

$$x=-6\sqrt{6}\approx -14.696938457$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply x^{2} by 1+2\sqrt{2}.

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

Use the distributive property to multiply \sqrt{2} by 2x^{2}-6x\sqrt{3}+3\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 3 and 2 to get 6.

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

Subtract \sqrt{2}\left(2x^{2}-6x\sqrt{3}\right) from both sides.

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

Subtract 6 from both sides.

$$6+x^{2}+2x^{2}\sqrt{2}-\sqrt{2}\left(2x^{2}-6x\sqrt{3}\right)-6=0$$

Use the distributive property to multiply -\sqrt{2} by 2x^{2}-6x\sqrt{3}.

$$6+x^{2}+2x^{2}\sqrt{2}-2x^{2}\sqrt{2}+6\sqrt{3}x\sqrt{2}-6=0$$

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

$$6+x^{2}+2x^{2}\sqrt{2}-2x^{2}\sqrt{2}+6\sqrt{6}x-6=0$$

Combine 2x^{2}\sqrt{2} and -2x^{2}\sqrt{2} to get 0.

$$6+x^{2}+6\sqrt{6}x-6=0$$

Subtract 6 from 6 to get 0.

$$x^{2}+6\sqrt{6}x=0$$

Factor out x.

$$x\left(x+6\sqrt{6}\right)=0$$

To find equation solutions, solve x=0 and x+6\sqrt{6}=0.

$$x=0 x=-6\sqrt{6}$$