Solve 1/3*(1,5a-b)^2-3/4*(1/3b+a)^2

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

Evaluate

$$-\frac{b\left(6a-b\right)}{4}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(1,5a-b\right)^{2}.

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

Use the distributive property to multiply \frac{1}{3} by 2,25a^{2}-3ab+b^{2}.

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

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

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

Use the distributive property to multiply -\frac{3}{4} by \frac{1}{9}b^{2}+\frac{2}{3}ba+a^{2}.

$$\frac{3}{4}a^{2}-ab+\frac{1}{3}b^{2}-\frac{1}{12}b^{2}-\frac{1}{2}ba-\frac{3}{4}a^{2}$$

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

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

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

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

Combine \frac{3}{4}a^{2} and -\frac{3}{4}a^{2} to get 0.

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

Expand

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

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(1,5a-b\right)^{2}.

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

Use the distributive property to multiply \frac{1}{3} by 2,25a^{2}-3ab+b^{2}.

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

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

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

Use the distributive property to multiply -\frac{3}{4} by \frac{1}{9}b^{2}+\frac{2}{3}ba+a^{2}.

$$\frac{3}{4}a^{2}-ab+\frac{1}{3}b^{2}-\frac{1}{12}b^{2}-\frac{1}{2}ba-\frac{3}{4}a^{2}$$

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

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

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

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

Combine \frac{3}{4}a^{2} and -\frac{3}{4}a^{2} to get 0.

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