Solve (frac{2sqrt{3}x}{3})^2=x(sqrt{7}-x)

$$( \frac { 2 \sqrt { 3 } x } { 3 } ) ^ { 2 } = x ( \sqrt { 7 } - x )$$

Solve for x

$$x = \frac{3 \sqrt{7}}{7} \approx 1.133893419$$

$$x=0$$

Steps Using Factoring

To raise \frac{2\sqrt{3}x}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(2\sqrt{3}x\right)^{2}}{3^{2}}=x\left(\sqrt{7}-x\right)$$

Use the distributive property to multiply x by \sqrt{7}-x.

$$\frac{\left(2\sqrt{3}x\right)^{2}}{3^{2}}=x\sqrt{7}-x^{2}$$

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

$$\frac{2^{2}\left(\sqrt{3}\right)^{2}x^{2}}{3^{2}}=x\sqrt{7}-x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{4\left(\sqrt{3}\right)^{2}x^{2}}{3^{2}}=x\sqrt{7}-x^{2}$$

The square of \sqrt{3} is 3.

$$\frac{4\times 3x^{2}}{3^{2}}=x\sqrt{7}-x^{2}$$

Multiply 4 and 3 to get 12.

$$\frac{12x^{2}}{3^{2}}=x\sqrt{7}-x^{2}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{12x^{2}}{9}=x\sqrt{7}-x^{2}$$

Divide 12x^{2} by 9 to get \frac{4}{3}x^{2}.

$$\frac{4}{3}x^{2}=x\sqrt{7}-x^{2}$$

Subtract x\sqrt{7} from both sides.

$$\frac{4}{3}x^{2}-x\sqrt{7}=-x^{2}$$

Add x^{2} to both sides.

$$\frac{4}{3}x^{2}-x\sqrt{7}+x^{2}=0$$

Combine \frac{4}{3}x^{2} and x^{2} to get \frac{7}{3}x^{2}.

$$\frac{7}{3}x^{2}-x\sqrt{7}=0$$

Factor out x.

$$x\left(\frac{7}{3}x-\sqrt{7}\right)=0$$

To find equation solutions, solve x=0 and \frac{7x}{3}-\sqrt{7}=0.

$$x=0 x=\frac{3\sqrt{7}}{7}$$