Solve (1/2a+2/3b)(3/2a-1/2b)-(frac{1}{2}a+b)(2a-frac{1}{2}b)

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

Evaluate

$$-ab+\frac{b^{2}}{6}-\frac{a^{2}}{4}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}a+\frac{2}{3}b by each term of \frac{3}{2}a-\frac{1}{2}b.

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

Multiply a and a to get a^{2}.

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

Multiply b and b to get b^{2}.

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

Multiply \frac{1}{2} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 3}{2\times 2}.

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

Multiply \frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Cancel out \frac{2}{3} and its reciprocal \frac{3}{2}.

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

Combine -\frac{1}{4}ab and ba to get \frac{3}{4}ab.

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

Multiply \frac{2}{3} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

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

Apply the distributive property by multiplying each term of \frac{1}{2}a+b by each term of 2a-\frac{1}{2}b.

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

Multiply a and a to get a^{2}.

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

Multiply b and b to get b^{2}.

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

Cancel out 2 and 2.

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

Multiply \frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Combine -\frac{1}{4}ab and 2ba to get \frac{7}{4}ab.

$$\frac{3}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-\left(a^{2}+\frac{7}{4}ab+b^{2}\left(-\frac{1}{2}\right)\right)$$

To find the opposite of a^{2}+\frac{7}{4}ab+b^{2}\left(-\frac{1}{2}\right), find the opposite of each term.

$$\frac{3}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-a^{2}-\frac{7}{4}ab-b^{2}\left(-\frac{1}{2}\right)$$

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

$$-\frac{1}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-\frac{7}{4}ab-b^{2}\left(-\frac{1}{2}\right)$$

Combine \frac{3}{4}ab and -\frac{7}{4}ab to get -ab.

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

Multiply -1 and -\frac{1}{2} to get \frac{1}{2}.

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

Combine -\frac{1}{3}b^{2} and \frac{1}{2}b^{2} to get \frac{1}{6}b^{2}.

$$-\frac{1}{4}a^{2}-ab+\frac{1}{6}b^{2}$$

Expand

$$-ab+\frac{b^{2}}{6}-\frac{a^{2}}{4}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}a+\frac{2}{3}b by each term of \frac{3}{2}a-\frac{1}{2}b.

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

Multiply a and a to get a^{2}.

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

Multiply b and b to get b^{2}.

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

Multiply \frac{1}{2} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\times 3}{2\times 2}.

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

Multiply \frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Cancel out \frac{2}{3} and its reciprocal \frac{3}{2}.

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

Combine -\frac{1}{4}ab and ba to get \frac{3}{4}ab.

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

Multiply \frac{2}{3} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

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

Apply the distributive property by multiplying each term of \frac{1}{2}a+b by each term of 2a-\frac{1}{2}b.

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

Multiply a and a to get a^{2}.

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

Multiply b and b to get b^{2}.

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

Cancel out 2 and 2.

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

Multiply \frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Combine -\frac{1}{4}ab and 2ba to get \frac{7}{4}ab.

$$\frac{3}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-\left(a^{2}+\frac{7}{4}ab+b^{2}\left(-\frac{1}{2}\right)\right)$$

To find the opposite of a^{2}+\frac{7}{4}ab+b^{2}\left(-\frac{1}{2}\right), find the opposite of each term.

$$\frac{3}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-a^{2}-\frac{7}{4}ab-b^{2}\left(-\frac{1}{2}\right)$$

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

$$-\frac{1}{4}a^{2}+\frac{3}{4}ab-\frac{1}{3}b^{2}-\frac{7}{4}ab-b^{2}\left(-\frac{1}{2}\right)$$

Combine \frac{3}{4}ab and -\frac{7}{4}ab to get -ab.

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

Multiply -1 and -\frac{1}{2} to get \frac{1}{2}.

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

Combine -\frac{1}{3}b^{2} and \frac{1}{2}b^{2} to get \frac{1}{6}b^{2}.

$$-\frac{1}{4}a^{2}-ab+\frac{1}{6}b^{2}$$