Solve (4/3a^2-ab-b^2)-(a^2-3/2ab+1/2b^2)

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

Evaluate

$$\frac{\left(2a-3b\right)\left(a+3b\right)}{6}$$

Solution Steps

To find the opposite of a^{2}-\frac{3}{2}ab+\frac{1}{2}b^{2}, find the opposite of each term.

$$\frac{4}{3}a^{2}-ab-b^{2}-a^{2}+\frac{3}{2}ab-\frac{1}{2}b^{2}$$

Combine \frac{4}{3}a^{2} and -a^{2} to get \frac{1}{3}a^{2}.

$$\frac{1}{3}a^{2}-ab-b^{2}+\frac{3}{2}ab-\frac{1}{2}b^{2}$$

Combine -ab and \frac{3}{2}ab to get \frac{1}{2}ab.

$$\frac{1}{3}a^{2}+\frac{1}{2}ab-b^{2}-\frac{1}{2}b^{2}$$

Combine -b^{2} and -\frac{1}{2}b^{2} to get -\frac{3}{2}b^{2}.

$$\frac{1}{3}a^{2}+\frac{1}{2}ab-\frac{3}{2}b^{2}$$

Expand

$$\frac{ab}{2}+\frac{a^{2}}{3}-\frac{3b^{2}}{2}$$

Solution Steps

To find the opposite of a^{2}-\frac{3}{2}ab+\frac{1}{2}b^{2}, find the opposite of each term.

$$\frac{4}{3}a^{2}-ab-b^{2}-a^{2}+\frac{3}{2}ab-\frac{1}{2}b^{2}$$

Combine \frac{4}{3}a^{2} and -a^{2} to get \frac{1}{3}a^{2}.

$$\frac{1}{3}a^{2}-ab-b^{2}+\frac{3}{2}ab-\frac{1}{2}b^{2}$$

Combine -ab and \frac{3}{2}ab to get \frac{1}{2}ab.

$$\frac{1}{3}a^{2}+\frac{1}{2}ab-b^{2}-\frac{1}{2}b^{2}$$

Combine -b^{2} and -\frac{1}{2}b^{2} to get -\frac{3}{2}b^{2}.

$$\frac{1}{3}a^{2}+\frac{1}{2}ab-\frac{3}{2}b^{2}$$