Solve 3^2+(3x)^2=(3+sqrt{3}x)^2

$$3 ^ { 2 } + ( 3 x ) ^ { 2 } = ( 3 + \sqrt { 3 } x ) ^ { 2 }$$

Solve for x

$$x=\sqrt{3}\approx 1.732050808$$

$$x=0$$

Steps Using Factoring

Calculate 3 to the power of 2 and get 9.

$$9+\left(3x\right)^{2}=\left(3+\sqrt{3}x\right)^{2}$$

Expand \left(3x\right)^{2}.

$$9+3^{2}x^{2}=\left(3+\sqrt{3}x\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9+9x^{2}=\left(3+\sqrt{3}x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{3}x\right)^{2}.

$$9+9x^{2}=9+6\sqrt{3}x+\left(\sqrt{3}\right)^{2}x^{2}$$

The square of \sqrt{3} is 3.

$$9+9x^{2}=9+6\sqrt{3}x+3x^{2}$$

Subtract 9 from both sides.

$$9+9x^{2}-9=6\sqrt{3}x+3x^{2}$$

Subtract 9 from 9 to get 0.

$$9x^{2}=6\sqrt{3}x+3x^{2}$$

Subtract 6\sqrt{3}x from both sides.

$$9x^{2}-6\sqrt{3}x=3x^{2}$$

Subtract 3x^{2} from both sides.

$$9x^{2}-6\sqrt{3}x-3x^{2}=0$$

Combine 9x^{2} and -3x^{2} to get 6x^{2}.

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

Factor out x.

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

To find equation solutions, solve x=0 and 6x-6\sqrt{3}=0.

$$x=0 x=\sqrt{3}$$