Solve 1/2(3-2mu)^2+1/2(6-mu)^2+1/2mu^2=

$$\frac { 1 } { 2 } ( 3 - 2 \mu ) ^ { 2 } + \frac { 1 } { 2 } ( 6 - \mu ) ^ { 2 } + \frac { 1 } { 2 } \mu ^ { 2 } =$$

Evaluate

$$3\mu ^{2}-12\mu +\frac{45}{2}$$

Solution Steps

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

$$\frac{1}{2}\left(9-12\mu +4\mu ^{2}\right)+\frac{1}{2}\left(6-\mu \right)^{2}+\frac{1}{2}\mu ^{2}$$

Use the distributive property to multiply \frac{1}{2} by 9-12\mu +4\mu ^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+\frac{1}{2}\left(6-\mu \right)^{2}+\frac{1}{2}\mu ^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-\mu \right)^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+\frac{1}{2}\left(36-12\mu +\mu ^{2}\right)+\frac{1}{2}\mu ^{2}$$

Use the distributive property to multiply \frac{1}{2} by 36-12\mu +\mu ^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+18-6\mu +\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Add \frac{9}{2} and 18 to get \frac{45}{2}.

$$\frac{45}{2}-6\mu +2\mu ^{2}-6\mu +\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine -6\mu and -6\mu to get -12\mu .

$$\frac{45}{2}-12\mu +2\mu ^{2}+\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine 2\mu ^{2} and \frac{1}{2}\mu ^{2} to get \frac{5}{2}\mu ^{2}.

$$\frac{45}{2}-12\mu +\frac{5}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine \frac{5}{2}\mu ^{2} and \frac{1}{2}\mu ^{2} to get 3\mu ^{2}.

$$\frac{45}{2}-12\mu +3\mu ^{2}$$

Expand

$$3\mu ^{2}-12\mu +\frac{45}{2}$$

Solution Steps

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

$$\frac{1}{2}\left(9-12\mu +4\mu ^{2}\right)+\frac{1}{2}\left(6-\mu \right)^{2}+\frac{1}{2}\mu ^{2}$$

Use the distributive property to multiply \frac{1}{2} by 9-12\mu +4\mu ^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+\frac{1}{2}\left(6-\mu \right)^{2}+\frac{1}{2}\mu ^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-\mu \right)^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+\frac{1}{2}\left(36-12\mu +\mu ^{2}\right)+\frac{1}{2}\mu ^{2}$$

Use the distributive property to multiply \frac{1}{2} by 36-12\mu +\mu ^{2}.

$$\frac{9}{2}-6\mu +2\mu ^{2}+18-6\mu +\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Add \frac{9}{2} and 18 to get \frac{45}{2}.

$$\frac{45}{2}-6\mu +2\mu ^{2}-6\mu +\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine -6\mu and -6\mu to get -12\mu .

$$\frac{45}{2}-12\mu +2\mu ^{2}+\frac{1}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine 2\mu ^{2} and \frac{1}{2}\mu ^{2} to get \frac{5}{2}\mu ^{2}.

$$\frac{45}{2}-12\mu +\frac{5}{2}\mu ^{2}+\frac{1}{2}\mu ^{2}$$

Combine \frac{5}{2}\mu ^{2} and \frac{1}{2}\mu ^{2} to get 3\mu ^{2}.

$$\frac{45}{2}-12\mu +3\mu ^{2}$$