Solve (2/3a+1/2b-3/5c)*(2/3a+frac{1}{2}b+frac{3}{5}c)+(-frac{2}{3}a-b)*frac{2}{3}a-frac{1}{2}*[frac{1}{2}b^2-(+frac{4}{5}c)^2]

$$( \frac { 2 } { 3 } a + \frac { 1 } { 2 } b - \frac { 3 } { 5 } c ) \cdot ( \frac { 2 } { 3 } a + \frac { 1 } { 2 } b + \frac { 3 } { 5 } c ) + ( - \frac { 2 } { 3 } a - b ) \cdot \frac { 2 } { 3 } a - \frac { 1 } { 2 } \cdot [ \frac { 1 } { 2 } b ^ { 2 } - ( + \frac { 4 } { 5 } c ) ^ { 2 } ]$$

Evaluate

$$-\frac{c^{2}}{25}$$

Solution Steps

Use the distributive property to multiply \frac{2}{3}a+\frac{1}{2}b-\frac{3}{5}c by \frac{2}{3}a+\frac{1}{2}b+\frac{3}{5}c and combine like terms.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}+\left(-\frac{2}{3}a-b\right)\times \frac{2}{3}a-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Use the distributive property to multiply -\frac{2}{3}a-b by \frac{2}{3}.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}+\left(-\frac{4}{9}a-\frac{2}{3}b\right)a-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Use the distributive property to multiply -\frac{4}{9}a-\frac{2}{3}b by a.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{4}{9}a^{2}-\frac{2}{3}ba-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Combine \frac{4}{9}a^{2} and -\frac{4}{9}a^{2} to get 0.

$$\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{2}{3}ba-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Combine \frac{2}{3}ab and -\frac{2}{3}ba to get 0.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Expand \left(\frac{4}{5}c\right)^{2}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}\right)^{2}c^{2}\right)$$

Calculate \frac{4}{5} to the power of 2 and get \frac{16}{25}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\frac{16}{25}c^{2}\right)$$

Use the distributive property to multiply -\frac{1}{2} by \frac{1}{2}b^{2}-\frac{16}{25}c^{2}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{4}b^{2}+\frac{8}{25}c^{2}$$

Combine \frac{1}{4}b^{2} and -\frac{1}{4}b^{2} to get 0.

$$-\frac{9}{25}c^{2}+\frac{8}{25}c^{2}$$

Combine -\frac{9}{25}c^{2} and \frac{8}{25}c^{2} to get -\frac{1}{25}c^{2}.

$$-\frac{1}{25}c^{2}$$

Expand

$$-\frac{c^{2}}{25}$$

Solution Steps

Use the distributive property to multiply \frac{2}{3}a+\frac{1}{2}b-\frac{3}{5}c by \frac{2}{3}a+\frac{1}{2}b+\frac{3}{5}c and combine like terms.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}+\left(-\frac{2}{3}a-b\right)\times \frac{2}{3}a-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Use the distributive property to multiply -\frac{2}{3}a-b by \frac{2}{3}.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}+\left(-\frac{4}{9}a-\frac{2}{3}b\right)a-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Use the distributive property to multiply -\frac{4}{9}a-\frac{2}{3}b by a.

$$\frac{4}{9}a^{2}+\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{4}{9}a^{2}-\frac{2}{3}ba-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Combine \frac{4}{9}a^{2} and -\frac{4}{9}a^{2} to get 0.

$$\frac{2}{3}ab+\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{2}{3}ba-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Combine \frac{2}{3}ab and -\frac{2}{3}ba to get 0.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}c\right)^{2}\right)$$

Expand \left(\frac{4}{5}c\right)^{2}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\left(\frac{4}{5}\right)^{2}c^{2}\right)$$

Calculate \frac{4}{5} to the power of 2 and get \frac{16}{25}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{2}\left(\frac{1}{2}b^{2}-\frac{16}{25}c^{2}\right)$$

Use the distributive property to multiply -\frac{1}{2} by \frac{1}{2}b^{2}-\frac{16}{25}c^{2}.

$$\frac{1}{4}b^{2}-\frac{9}{25}c^{2}-\frac{1}{4}b^{2}+\frac{8}{25}c^{2}$$

Combine \frac{1}{4}b^{2} and -\frac{1}{4}b^{2} to get 0.

$$-\frac{9}{25}c^{2}+\frac{8}{25}c^{2}$$

Combine -\frac{9}{25}c^{2} and \frac{8}{25}c^{2} to get -\frac{1}{25}c^{2}.

$$-\frac{1}{25}c^{2}$$