Solve 5^2=x^2+(sqrt{3}+frac{2sqrt{3}}{3})^2

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

Solve for x

$$x = \frac{5 \sqrt{6}}{3} \approx 4.082482905$$

$$x = -\frac{5 \sqrt{6}}{3} \approx -4.082482905$$

Steps by Finding Square Root

Calculate 5 to the power of 2 and get 25.

$$25=x^{2}+\left(\sqrt{3}+\frac{2\sqrt{3}}{3}\right)^{2}$$

Combine \sqrt{3} and \frac{2\sqrt{3}}{3} to get \frac{5}{3}\sqrt{3}.

$$25=x^{2}+\left(\frac{5}{3}\sqrt{3}\right)^{2}$$

Expand \left(\frac{5}{3}\sqrt{3}\right)^{2}.

$$25=x^{2}+\left(\frac{5}{3}\right)^{2}\left(\sqrt{3}\right)^{2}$$

Calculate \frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$25=x^{2}+\frac{25}{9}\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$25=x^{2}+\frac{25}{9}\times 3$$

Multiply \frac{25}{9} and 3 to get \frac{25}{3}.

$$25=x^{2}+\frac{25}{3}$$

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

$$x^{2}+\frac{25}{3}=25$$

Subtract \frac{25}{3} from both sides.

$$x^{2}=25-\frac{25}{3}$$

Subtract \frac{25}{3} from 25 to get \frac{50}{3}.

$$x^{2}=\frac{50}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{5\sqrt{6}}{3} x=-\frac{5\sqrt{6}}{3}$$

Steps Using the Quadratic Formula

Calculate 5 to the power of 2 and get 25.

$$25=x^{2}+\left(\sqrt{3}+\frac{2\sqrt{3}}{3}\right)^{2}$$

Combine \sqrt{3} and \frac{2\sqrt{3}}{3} to get \frac{5}{3}\sqrt{3}.

$$25=x^{2}+\left(\frac{5}{3}\sqrt{3}\right)^{2}$$

Expand \left(\frac{5}{3}\sqrt{3}\right)^{2}.

$$25=x^{2}+\left(\frac{5}{3}\right)^{2}\left(\sqrt{3}\right)^{2}$$

Calculate \frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$25=x^{2}+\frac{25}{9}\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$25=x^{2}+\frac{25}{9}\times 3$$

Multiply \frac{25}{9} and 3 to get \frac{25}{3}.

$$25=x^{2}+\frac{25}{3}$$

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

$$x^{2}+\frac{25}{3}=25$$

Subtract 25 from both sides.

$$x^{2}+\frac{25}{3}-25=0$$

Subtract 25 from \frac{25}{3} to get -\frac{50}{3}.

$$x^{2}-\frac{50}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{50}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{50}{3}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{50}{3}\right)}}{2}$$

Multiply -4 times -\frac{50}{3}.

$$x=\frac{0±\sqrt{\frac{200}{3}}}{2}$$

Take the square root of \frac{200}{3}.

$$x=\frac{0±\frac{10\sqrt{6}}{3}}{2}$$

Now solve the equation x=\frac{0±\frac{10\sqrt{6}}{3}}{2} when ± is plus.

$$x=\frac{5\sqrt{6}}{3}$$

Now solve the equation x=\frac{0±\frac{10\sqrt{6}}{3}}{2} when ± is minus.

$$x=-\frac{5\sqrt{6}}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{6}}{3} x=-\frac{5\sqrt{6}}{3}$$