Solve x^2=(frac{2sqrt{6}r}{3}-x)^2+(frac{2sqrt{3}r}{3})^2

$$x ^ { 2 } = ( \frac { 2 \sqrt { 6 } r } { 3 } - x ) ^ { 2 } + ( \frac { 2 \sqrt { 3 } r } { 3 } ) ^ { 2 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{6}r}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&r=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{2\sqrt{6}r}{3}-x\right)^{2}.

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

To raise \frac{2\sqrt{6}r}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Express -2\times \frac{2\sqrt{6}r}{3} as a single fraction.

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

Express \frac{-2\times 2\sqrt{6}r}{3}x as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{-2\times 2\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}}{9}+\frac{3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9}+x^{2}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}}{9} and \frac{3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9}+x^{2}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(2\sqrt{6}r\right)^{2}}{3^{2}} and \frac{x^{2}\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Multiply -2 and 2 to get -4.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

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

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{-4\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{9}+\frac{3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{9} and \frac{3\left(-4\right)\sqrt{6}rx}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}+3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}} and \frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}+\left(2\sqrt{3}r\right)^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 3^{2} is 9. Multiply \frac{-4\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{9}$$

Since \frac{3\left(-4\right)\sqrt{6}rx}{9} and \frac{\left(2\sqrt{3}r\right)^{2}}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Expand \left(2\sqrt{6}r\right)^{2}.

$$x^{2}=\frac{2^{2}\left(\sqrt{6}\right)^{2}r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=\frac{4\left(\sqrt{6}\right)^{2}r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

The square of \sqrt{6} is 6.

$$x^{2}=\frac{4\times 6r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Multiply 4 and 6 to get 24.

$$x^{2}=\frac{24r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{24r^{2}+x^{2}\times 9}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{24r^{2}+x^{2}\times 9}{9}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Divide each term of 24r^{2}+x^{2}\times 9 by 9 to get \frac{8}{3}r^{2}+x^{2}.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Multiply 3 and -4 to get -12.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

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

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+2^{2}\left(\sqrt{3}\right)^{2}r^{2}}{9}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+4\left(\sqrt{3}\right)^{2}r^{2}}{9}$$

The square of \sqrt{3} is 3.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+4\times 3r^{2}}{9}$$

Multiply 4 and 3 to get 12.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

Subtract x^{2} from both sides.

$$x^{2}-x^{2}=\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

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

$$0=\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}=0$$

Subtract \frac{8}{3}r^{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{-12\sqrt{6}rx+12r^{2}}{9}=-\frac{8}{3}r^{2}$$

Multiply both sides of the equation by 9, the least common multiple of 9,3.

$$-12\sqrt{6}rx+12r^{2}=-24r^{2}$$

Subtract 12r^{2} from both sides.

$$-12\sqrt{6}rx=-24r^{2}-12r^{2}$$

Combine -24r^{2} and -12r^{2} to get -36r^{2}.

$$-12\sqrt{6}rx=-36r^{2}$$

The equation is in standard form.

$$\left(-12\sqrt{6}r\right)x=-36r^{2}$$

Divide both sides by -12\sqrt{6}r.

$$\frac{\left(-12\sqrt{6}r\right)x}{-12\sqrt{6}r}=-\frac{36r^{2}}{-12\sqrt{6}r}$$

Dividing by -12\sqrt{6}r undoes the multiplication by -12\sqrt{6}r.

$$x=-\frac{36r^{2}}{-12\sqrt{6}r}$$

Divide -36r^{2} by -12\sqrt{6}r.

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

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{6}r}{2}\text{, }&r\neq 0\\x\in \mathrm{R}\text{, }&r=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{2\sqrt{6}r}{3}-x\right)^{2}.

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

To raise \frac{2\sqrt{6}r}{3} to a power, raise both numerator and denominator to the power and then divide.

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

Express -2\times \frac{2\sqrt{6}r}{3} as a single fraction.

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

Express \frac{-2\times 2\sqrt{6}r}{3}x as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{-2\times 2\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}}{9}+\frac{3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9}+x^{2}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}}{9} and \frac{3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+3\left(-1\right)\times 2\times 2\sqrt{6}rx}{9}+x^{2}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(2\sqrt{6}r\right)^{2}}{3^{2}} and \frac{x^{2}\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Multiply -2 and 2 to get -4.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}+\left(\frac{2\sqrt{3}r}{3}\right)^{2}$$

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

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3^{2} and 3 is 9. Multiply \frac{-4\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{9}+\frac{3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{9} and \frac{3\left(-4\right)\sqrt{6}rx}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}+3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}}$$

Since \frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}} and \frac{\left(2\sqrt{3}r\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}+\left(2\sqrt{3}r\right)^{2}}{3^{2}}+\frac{-4\sqrt{6}rx}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 3^{2} is 9. Multiply \frac{-4\sqrt{6}rx}{3} times \frac{3}{3}.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx}{9}+\frac{\left(2\sqrt{3}r\right)^{2}}{9}$$

Since \frac{3\left(-4\right)\sqrt{6}rx}{9} and \frac{\left(2\sqrt{3}r\right)^{2}}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(2\sqrt{6}r\right)^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Expand \left(2\sqrt{6}r\right)^{2}.

$$x^{2}=\frac{2^{2}\left(\sqrt{6}\right)^{2}r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=\frac{4\left(\sqrt{6}\right)^{2}r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

The square of \sqrt{6} is 6.

$$x^{2}=\frac{4\times 6r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Multiply 4 and 6 to get 24.

$$x^{2}=\frac{24r^{2}+x^{2}\times 3^{2}}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{24r^{2}+x^{2}\times 9}{3^{2}}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{24r^{2}+x^{2}\times 9}{9}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Divide each term of 24r^{2}+x^{2}\times 9 by 9 to get \frac{8}{3}r^{2}+x^{2}.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{3\left(-4\right)\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

Multiply 3 and -4 to get -12.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+\left(2\sqrt{3}r\right)^{2}}{9}$$

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

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+2^{2}\left(\sqrt{3}\right)^{2}r^{2}}{9}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+4\left(\sqrt{3}\right)^{2}r^{2}}{9}$$

The square of \sqrt{3} is 3.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+4\times 3r^{2}}{9}$$

Multiply 4 and 3 to get 12.

$$x^{2}=\frac{8}{3}r^{2}+x^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

Subtract x^{2} from both sides.

$$x^{2}-x^{2}=\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

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

$$0=\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{8}{3}r^{2}+\frac{-12\sqrt{6}rx+12r^{2}}{9}=0$$

Subtract \frac{8}{3}r^{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{-12\sqrt{6}rx+12r^{2}}{9}=-\frac{8}{3}r^{2}$$

Multiply both sides of the equation by 9, the least common multiple of 9,3.

$$-12\sqrt{6}rx+12r^{2}=-24r^{2}$$

Subtract 12r^{2} from both sides.

$$-12\sqrt{6}rx=-24r^{2}-12r^{2}$$

Combine -24r^{2} and -12r^{2} to get -36r^{2}.

$$-12\sqrt{6}rx=-36r^{2}$$

The equation is in standard form.

$$\left(-12\sqrt{6}r\right)x=-36r^{2}$$

Divide both sides by -12\sqrt{6}r.

$$\frac{\left(-12\sqrt{6}r\right)x}{-12\sqrt{6}r}=-\frac{36r^{2}}{-12\sqrt{6}r}$$

Dividing by -12\sqrt{6}r undoes the multiplication by -12\sqrt{6}r.

$$x=-\frac{36r^{2}}{-12\sqrt{6}r}$$

Divide -36r^{2} by -12\sqrt{6}r.

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

Solve for r

$$r=\frac{\sqrt{6}x}{3}$$

$$r=0$$