Solve operatorname{van}(x)=E[(x-mu)^2]

$$\operatorname { van } ( x ) = E [ ( x - \mu ) ^ { 2 } ]$$

Solve for E (complex solution)

$$\left\{\begin{matrix}E=\frac{anvx}{\left(x-\mu \right)^{2}}\text{, }&x\neq \mu \\E\in \mathrm{C}\text{, }&\left(v=0\text{ and }x=\mu \right)\text{ or }\left(a=0\text{ and }x=\mu \right)\text{ or }\left(n=0\text{ and }x=\mu \right)\text{ or }\left(x=0\text{ and }\mu =0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$vanx=E\left(x^{2}-2x\mu +\mu ^{2}\right)$$

Use the distributive property to multiply E by x^{2}-2x\mu +\mu ^{2}.

$$vanx=Ex^{2}-2Ex\mu +E\mu ^{2}$$

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

$$Ex^{2}-2Ex\mu +E\mu ^{2}=vanx$$

Combine all terms containing E.

$$\left(x^{2}-2x\mu +\mu ^{2}\right)E=vanx$$

The equation is in standard form.

$$\left(x^{2}-2x\mu +\mu ^{2}\right)E=anvx$$

Divide both sides by x^{2}-2x\mu +\mu ^{2}.

$$\frac{\left(x^{2}-2x\mu +\mu ^{2}\right)E}{x^{2}-2x\mu +\mu ^{2}}=\frac{anvx}{x^{2}-2x\mu +\mu ^{2}}$$

Dividing by x^{2}-2x\mu +\mu ^{2} undoes the multiplication by x^{2}-2x\mu +\mu ^{2}.

$$E=\frac{anvx}{x^{2}-2x\mu +\mu ^{2}}$$

Divide vanx by x^{2}-2x\mu +\mu ^{2}.

$$E=\frac{anvx}{\left(x-\mu \right)^{2}}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{E\left(x-\mu \right)^{2}}{nvx}\text{, }&x\neq 0\text{ and }n\neq 0\text{ and }v\neq 0\\a\in \mathrm{C}\text{, }&\left(E=0\text{ and }n=0\right)\text{ or }\left(E=0\text{ and }x=0\right)\text{ or }\left(n=0\text{ and }x=\mu \right)\text{ or }\left(x=0\text{ and }\mu =0\right)\text{ or }\left(x=\mu \text{ and }v=0\text{ and }\mu \neq 0\text{ and }n\neq 0\right)\text{ or }\left(E=0\text{ and }v=0\text{ and }x\neq 0\text{ and }n\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$vanx=E\left(x^{2}-2x\mu +\mu ^{2}\right)$$

Use the distributive property to multiply E by x^{2}-2x\mu +\mu ^{2}.

$$vanx=Ex^{2}-2Ex\mu +E\mu ^{2}$$

The equation is in standard form.

$$nvxa=Ex^{2}-2Ex\mu +E\mu ^{2}$$

Divide both sides by vnx.

$$\frac{nvxa}{nvx}=\frac{E\left(x-\mu \right)^{2}}{nvx}$$

Dividing by vnx undoes the multiplication by vnx.

$$a=\frac{E\left(x-\mu \right)^{2}}{nvx}$$

Solve for E

$$\left\{\begin{matrix}E=\frac{anvx}{\left(x-\mu \right)^{2}}\text{, }&x\neq \mu \\E\in \mathrm{R}\text{, }&\left(v=0\text{ and }x=\mu \right)\text{ or }\left(a=0\text{ and }x=\mu \right)\text{ or }\left(n=0\text{ and }x=\mu \right)\text{ or }\left(x=0\text{ and }\mu =0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$vanx=E\left(x^{2}-2x\mu +\mu ^{2}\right)$$

Use the distributive property to multiply E by x^{2}-2x\mu +\mu ^{2}.

$$vanx=Ex^{2}-2Ex\mu +E\mu ^{2}$$

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

$$Ex^{2}-2Ex\mu +E\mu ^{2}=vanx$$

Combine all terms containing E.

$$\left(x^{2}-2x\mu +\mu ^{2}\right)E=vanx$$

The equation is in standard form.

$$\left(x^{2}-2x\mu +\mu ^{2}\right)E=anvx$$

Divide both sides by x^{2}-2x\mu +\mu ^{2}.

$$\frac{\left(x^{2}-2x\mu +\mu ^{2}\right)E}{x^{2}-2x\mu +\mu ^{2}}=\frac{anvx}{x^{2}-2x\mu +\mu ^{2}}$$

Dividing by x^{2}-2x\mu +\mu ^{2} undoes the multiplication by x^{2}-2x\mu +\mu ^{2}.

$$E=\frac{anvx}{x^{2}-2x\mu +\mu ^{2}}$$

Divide vanx by x^{2}-2x\mu +\mu ^{2}.

$$E=\frac{anvx}{\left(x-\mu \right)^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{E\left(x-\mu \right)^{2}}{nvx}\text{, }&x\neq 0\text{ and }n\neq 0\text{ and }v\neq 0\\a\in \mathrm{R}\text{, }&\left(E=0\text{ and }n=0\right)\text{ or }\left(E=0\text{ and }x=0\right)\text{ or }\left(n=0\text{ and }x=\mu \right)\text{ or }\left(x=0\text{ and }\mu =0\right)\text{ or }\left(x=\mu \text{ and }v=0\text{ and }\mu \neq 0\text{ and }n\neq 0\right)\text{ or }\left(E=0\text{ and }v=0\text{ and }x\neq 0\text{ and }n\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$vanx=E\left(x^{2}-2x\mu +\mu ^{2}\right)$$

Use the distributive property to multiply E by x^{2}-2x\mu +\mu ^{2}.

$$vanx=Ex^{2}-2Ex\mu +E\mu ^{2}$$

The equation is in standard form.

$$nvxa=Ex^{2}-2Ex\mu +E\mu ^{2}$$

Divide both sides by vnx.

$$\frac{nvxa}{nvx}=\frac{E\left(x-\mu \right)^{2}}{nvx}$$

Dividing by vnx undoes the multiplication by vnx.

$$a=\frac{E\left(x-\mu \right)^{2}}{nvx}$$