Solve x^2=(x-8/3)^2+(frac{4sqrt{5}}{3})^2

$$x ^ { 2 } = ( x - \frac { 8 } { 3 } ) ^ { 2 } + ( \frac { 4 \sqrt { 5 } } { 3 } ) ^ { 2 }$$

Solve for x

$$x=3$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{8}{3}\right)^{2}.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64}{9}+\left(\frac{4\sqrt{5}}{3}\right)^{2}$$

To raise \frac{4\sqrt{5}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64}{9}+\frac{\left(4\sqrt{5}\right)^{2}}{3^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 3^{2}.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64}{9}+\frac{\left(4\sqrt{5}\right)^{2}}{9}$$

Since \frac{64}{9} and \frac{\left(4\sqrt{5}\right)^{2}}{9} have the same denominator, add them by adding their numerators.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64+\left(4\sqrt{5}\right)^{2}}{9}$$

Expand \left(4\sqrt{5}\right)^{2}.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64+4^{2}\left(\sqrt{5}\right)^{2}}{9}$$

Calculate 4 to the power of 2 and get 16.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64+16\left(\sqrt{5}\right)^{2}}{9}$$

The square of \sqrt{5} is 5.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64+16\times 5}{9}$$

Multiply 16 and 5 to get 80.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{64+80}{9}$$

Add 64 and 80 to get 144.

$$x^{2}=x^{2}-\frac{16}{3}x+\frac{144}{9}$$

Divide 144 by 9 to get 16.

$$x^{2}=x^{2}-\frac{16}{3}x+16$$

Subtract x^{2} from both sides.

$$x^{2}-x^{2}=-\frac{16}{3}x+16$$

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

$$0=-\frac{16}{3}x+16$$

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

$$-\frac{16}{3}x+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{16}{3}x=-16$$

Multiply both sides by -\frac{3}{16}, the reciprocal of -\frac{16}{3}.

$$x=-16\left(-\frac{3}{16}\right)$$

Multiply -16 and -\frac{3}{16} to get 3.

$$x=3$$