Solve (1/4x+2a)(1/3x+8a)-(x+2/3a)^2+(a-x)(2a+1/3x)+(frac{1}{3}a+frac{1}{2}x)^2+frac{1}{3}a^2

$$( \frac { 1 } { 4 } x + 2 a ) ( \frac { 1 } { 3 } x + 8 a ) - ( x + \frac { 2 } { 3 } a ) ^ { 2 } + ( a - x ) ( 2 a + \frac { 1 } { 3 } x ) + ( \frac { 1 } { 3 } a + \frac { 1 } { 2 } x ) ^ { 2 } + \frac { 1 } { 3 } a ^ { 2 }$$

Evaluate

$$18a^{2}-x^{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4}x+2a by \frac{1}{3}x+8a and combine like terms.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\left(x+\frac{2}{3}a\right)^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+\frac{2}{3}a\right)^{2}.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\left(x^{2}+\frac{4}{3}xa+\frac{4}{9}a^{2}\right)+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

To find the opposite of x^{2}+\frac{4}{3}xa+\frac{4}{9}a^{2}, find the opposite of each term.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-x^{2}-\frac{4}{3}xa-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{1}{12}x^{2} and -x^{2} to get -\frac{11}{12}x^{2}.

$$-\frac{11}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\frac{4}{3}xa-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{8}{3}xa and -\frac{4}{3}xa to get \frac{4}{3}xa.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+16a^{2}-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine 16a^{2} and -\frac{4}{9}a^{2} to get \frac{140}{9}a^{2}.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{140}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use the distributive property to multiply a-x by 2a+\frac{1}{3}x and combine like terms.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{140}{9}a^{2}+2a^{2}-\frac{5}{3}ax-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{140}{9}a^{2} and 2a^{2} to get \frac{158}{9}a^{2}.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{158}{9}a^{2}-\frac{5}{3}ax-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{4}{3}xa and -\frac{5}{3}ax to get -\frac{1}{3}xa.

$$-\frac{11}{12}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{11}{12}x^{2} and -\frac{1}{3}x^{2} to get -\frac{5}{4}x^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}+\frac{1}{9}a^{2}+\frac{1}{3}ax+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine \frac{158}{9}a^{2} and \frac{1}{9}a^{2} to get \frac{53}{3}a^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{53}{3}a^{2}+\frac{1}{3}ax+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{1}{3}xa and \frac{1}{3}ax to get 0.

$$-\frac{5}{4}x^{2}+\frac{53}{3}a^{2}+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{5}{4}x^{2} and \frac{1}{4}x^{2} to get -x^{2}.

$$-x^{2}+\frac{53}{3}a^{2}+\frac{1}{3}a^{2}$$

Combine \frac{53}{3}a^{2} and \frac{1}{3}a^{2} to get 18a^{2}.

$$-x^{2}+18a^{2}$$

Expand

$$18a^{2}-x^{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4}x+2a by \frac{1}{3}x+8a and combine like terms.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\left(x+\frac{2}{3}a\right)^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+\frac{2}{3}a\right)^{2}.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\left(x^{2}+\frac{4}{3}xa+\frac{4}{9}a^{2}\right)+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

To find the opposite of x^{2}+\frac{4}{3}xa+\frac{4}{9}a^{2}, find the opposite of each term.

$$\frac{1}{12}x^{2}+\frac{8}{3}xa+16a^{2}-x^{2}-\frac{4}{3}xa-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{1}{12}x^{2} and -x^{2} to get -\frac{11}{12}x^{2}.

$$-\frac{11}{12}x^{2}+\frac{8}{3}xa+16a^{2}-\frac{4}{3}xa-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{8}{3}xa and -\frac{4}{3}xa to get \frac{4}{3}xa.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+16a^{2}-\frac{4}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine 16a^{2} and -\frac{4}{9}a^{2} to get \frac{140}{9}a^{2}.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{140}{9}a^{2}+\left(a-x\right)\left(2a+\frac{1}{3}x\right)+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use the distributive property to multiply a-x by 2a+\frac{1}{3}x and combine like terms.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{140}{9}a^{2}+2a^{2}-\frac{5}{3}ax-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{140}{9}a^{2} and 2a^{2} to get \frac{158}{9}a^{2}.

$$-\frac{11}{12}x^{2}+\frac{4}{3}xa+\frac{158}{9}a^{2}-\frac{5}{3}ax-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine \frac{4}{3}xa and -\frac{5}{3}ax to get -\frac{1}{3}xa.

$$-\frac{11}{12}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}-\frac{1}{3}x^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{11}{12}x^{2} and -\frac{1}{3}x^{2} to get -\frac{5}{4}x^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}+\left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}+\frac{1}{3}a^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(\frac{1}{3}a+\frac{1}{2}x\right)^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{158}{9}a^{2}+\frac{1}{9}a^{2}+\frac{1}{3}ax+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine \frac{158}{9}a^{2} and \frac{1}{9}a^{2} to get \frac{53}{3}a^{2}.

$$-\frac{5}{4}x^{2}-\frac{1}{3}xa+\frac{53}{3}a^{2}+\frac{1}{3}ax+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{1}{3}xa and \frac{1}{3}ax to get 0.

$$-\frac{5}{4}x^{2}+\frac{53}{3}a^{2}+\frac{1}{4}x^{2}+\frac{1}{3}a^{2}$$

Combine -\frac{5}{4}x^{2} and \frac{1}{4}x^{2} to get -x^{2}.

$$-x^{2}+\frac{53}{3}a^{2}+\frac{1}{3}a^{2}$$

Combine \frac{53}{3}a^{2} and \frac{1}{3}a^{2} to get 18a^{2}.

$$-x^{2}+18a^{2}$$