Solve sigma_x^2=(-2-0)^2*4/9+(0*0)^2*3/9+(1*9)^2*2/9

$$\sigma _ { x } ^ { 2 } = ( - 2 - 0 ) ^ { 2 } \times \frac { 4 } { 9 } + ( 0 \times 0 ) ^ { 2 } \times \frac { 3 } { 9 } + ( 1 \times 9 ) ^ { 2 } \times \frac { 2 } { 9 }$$

Solve for σ_x

$$\sigma _{x}=\frac{\sqrt{178}}{3}\approx 4.447221355$$

$$\sigma _{x}=-\frac{\sqrt{178}}{3}\approx -4.447221355$$

Steps by Finding Square Root

Subtract 0 from -2 to get -2.

$$\sigma _{x}^{2}=\left(-2\right)^{2}\times \frac{4}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Calculate -2 to the power of 2 and get 4.

$$\sigma _{x}^{2}=4\times \frac{4}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 4 and \frac{4}{9} to get \frac{16}{9}.

$$\sigma _{x}^{2}=\frac{16}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 0 and 0 to get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Calculate 0 to the power of 2 and get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$\sigma _{x}^{2}=\frac{16}{9}+0\times \frac{1}{3}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 0 and \frac{1}{3} to get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Add \frac{16}{9} and 0 to get \frac{16}{9}.

$$\sigma _{x}^{2}=\frac{16}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 1 and 9 to get 9.

$$\sigma _{x}^{2}=\frac{16}{9}+9^{2}\times \frac{2}{9}$$

Calculate 9 to the power of 2 and get 81.

$$\sigma _{x}^{2}=\frac{16}{9}+81\times \frac{2}{9}$$

Multiply 81 and \frac{2}{9} to get 18.

$$\sigma _{x}^{2}=\frac{16}{9}+18$$

Add \frac{16}{9} and 18 to get \frac{178}{9}.

$$\sigma _{x}^{2}=\frac{178}{9}$$

Take the square root of both sides of the equation.

$$\sigma _{x}=\frac{\sqrt{178}}{3} \sigma _{x}=-\frac{\sqrt{178}}{3}$$

Steps Using the Quadratic Formula

Subtract 0 from -2 to get -2.

$$\sigma _{x}^{2}=\left(-2\right)^{2}\times \frac{4}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Calculate -2 to the power of 2 and get 4.

$$\sigma _{x}^{2}=4\times \frac{4}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 4 and \frac{4}{9} to get \frac{16}{9}.

$$\sigma _{x}^{2}=\frac{16}{9}+\left(0\times 0\right)^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 0 and 0 to get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0^{2}\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Calculate 0 to the power of 2 and get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0\times \frac{3}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$\sigma _{x}^{2}=\frac{16}{9}+0\times \frac{1}{3}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 0 and \frac{1}{3} to get 0.

$$\sigma _{x}^{2}=\frac{16}{9}+0+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Add \frac{16}{9} and 0 to get \frac{16}{9}.

$$\sigma _{x}^{2}=\frac{16}{9}+\left(1\times 9\right)^{2}\times \frac{2}{9}$$

Multiply 1 and 9 to get 9.

$$\sigma _{x}^{2}=\frac{16}{9}+9^{2}\times \frac{2}{9}$$

Calculate 9 to the power of 2 and get 81.

$$\sigma _{x}^{2}=\frac{16}{9}+81\times \frac{2}{9}$$

Multiply 81 and \frac{2}{9} to get 18.

$$\sigma _{x}^{2}=\frac{16}{9}+18$$

Add \frac{16}{9} and 18 to get \frac{178}{9}.

$$\sigma _{x}^{2}=\frac{178}{9}$$

Subtract \frac{178}{9} from both sides.

$$\sigma _{x}^{2}-\frac{178}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{178}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\sigma _{x}=\frac{0±\sqrt{0^{2}-4\left(-\frac{178}{9}\right)}}{2}$$

Square 0.

$$\sigma _{x}=\frac{0±\sqrt{-4\left(-\frac{178}{9}\right)}}{2}$$

Multiply -4 times -\frac{178}{9}.

$$\sigma _{x}=\frac{0±\sqrt{\frac{712}{9}}}{2}$$

Take the square root of \frac{712}{9}.

$$\sigma _{x}=\frac{0±\frac{2\sqrt{178}}{3}}{2}$$

Now solve the equation \sigma _{x}=\frac{0±\frac{2\sqrt{178}}{3}}{2} when ± is plus.

$$\sigma _{x}=\frac{\sqrt{178}}{3}$$

Now solve the equation \sigma _{x}=\frac{0±\frac{2\sqrt{178}}{3}}{2} when ± is minus.

$$\sigma _{x}=-\frac{\sqrt{178}}{3}$$

The equation is now solved.

$$\sigma _{x}=\frac{\sqrt{178}}{3} \sigma _{x}=-\frac{\sqrt{178}}{3}$$