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

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

Evaluate

$$-b\left(3a+5b\right)$$

Solution Steps

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

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\left(-\frac{1}{2}a+3b\right)^{2}-2\left(-a\right)^{2}$$

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

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\left(\frac{1}{4}a^{2}-3ab+9b^{2}\right)-2\left(-a\right)^{2}$$

To find the opposite of \frac{1}{4}a^{2}-3ab+9b^{2}, find the opposite of each term.

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\frac{1}{4}a^{2}+3ab-9b^{2}-2\left(-a\right)^{2}$$

Combine \frac{9}{4}a^{2} and -\frac{1}{4}a^{2} to get 2a^{2}.

$$2a^{2}-6ab+4b^{2}+3ab-9b^{2}-2\left(-a\right)^{2}$$

Combine -6ab and 3ab to get -3ab.

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

Combine 4b^{2} and -9b^{2} to get -5b^{2}.

$$2a^{2}-3ab-5b^{2}-2\left(-a\right)^{2}$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}-3ab-5b^{2}-2a^{2}$$

Combine 2a^{2} and -2a^{2} to get 0.

$$-3ab-5b^{2}$$

Expand

$$-3ab-5b^{2}$$

Solution Steps

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

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\left(-\frac{1}{2}a+3b\right)^{2}-2\left(-a\right)^{2}$$

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

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\left(\frac{1}{4}a^{2}-3ab+9b^{2}\right)-2\left(-a\right)^{2}$$

To find the opposite of \frac{1}{4}a^{2}-3ab+9b^{2}, find the opposite of each term.

$$\frac{9}{4}a^{2}-6ab+4b^{2}-\frac{1}{4}a^{2}+3ab-9b^{2}-2\left(-a\right)^{2}$$

Combine \frac{9}{4}a^{2} and -\frac{1}{4}a^{2} to get 2a^{2}.

$$2a^{2}-6ab+4b^{2}+3ab-9b^{2}-2\left(-a\right)^{2}$$

Combine -6ab and 3ab to get -3ab.

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

Combine 4b^{2} and -9b^{2} to get -5b^{2}.

$$2a^{2}-3ab-5b^{2}-2\left(-a\right)^{2}$$

Calculate -a to the power of 2 and get a^{2}.

$$2a^{2}-3ab-5b^{2}-2a^{2}$$

Combine 2a^{2} and -2a^{2} to get 0.

$$-3ab-5b^{2}$$